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PLANE TRIGONOMETRY 
AND NUMERICAL COMPUTATION 



A SERIES OF MATHEMATICAL TEXTS 

EDITED BY 

EARLE RAYMOND HEDRICK 



THE CALCULUS 

By Ellery Williams Davis and William Charles 
Brenke. 

ANALYTIC GEOMETRY AND ALGEBRA 

By Alexander Ziwet and Louis Allen Hopkins. 

ELEMENTS OF ANALYTIC GEOMETRY 

By Alexander Ziwet and Louis Allen Hopkins. 

PLANE AND SPHERICAL TRIGONOMETRY WITH COM- 
PLETE TABLES 
By Alfred Monroe Kenyon and Louis Ingold. 

PLANE AND SPHERICAL TRIGONOMETRY WITH BRIEF 
TABLES 
By Alfred Monroe Kenyon and Louis Ingold. 

ELEMENTARY MATHEMATICAL ANALYSIS 

By John Wesley Young and Frank Millett Morgan. 

COLLEGE ALGEBRA 

By Ernest Brown Skinner. 

ELEMENTS OF PLANE TRIGONOMETRY WITH COMPLETE 
TABLES 
By Alfred Moore Kenyon and Louis Ingold. 

ELEMENTS OF PLANE TRIGONOMETRY WITH BRIEF 
TABLES 
By Alfred Moore Kenyon and Louis Ingold. 

THE MACMILLAN TABLES 

Prepared under the direction of Earle Raymond Hedrick. 

PLANE GEOMETRY 

By Walter Burton Ford and Charles Ammerman. 

PLANE AND SOLID GEOMETRY 

By Walter Burton Ford and Charles Ammerman. 

SOLID GEOMETRY 

By Walter Burton Ford and Charles Ammerman. 

CONSTRUCTIVE GEOMETRY 

Prepared under the direction of Earle Raymond Hedrick. 

JUNIOR HIGH SCHOOL MATHEMATICS 

By William Ledley Vosburgh and Frederick William 
Gentleman. 



PLANE TRIGONOMETRY 

AND NUMERICAL 

COMPUTATION 



BY 
JOHN WESLEY YOUNG 

PROFESSOR OF MATHEMATICS 
DARTMOUTH COLLEGE 

AND 

FRANK MILLETT MORGAN 

ASSISTANT PROFESSOR OF MATHEMATICS 
DARTMOUTH COLLEGE 



- Neto gorfe 
THE MACMILLAN COMPANY 
1919 

All rights reserved 






Copyright, 1919, 
By the MACMILLAN COMPANY. 



Set up and electrotyped. Published October, 1919. 



NOV 12 1919 



Nortoooti Press 

J. S. Gushing Co. — Berwick & Smith Co. 

Norwood, Mas3., U.S.A. 



©C1.A535687 



PREFACE 

Ever since the publication of our Elementary Mathematical 
Analysis (The Macmillan Co., 1917) we have been asked by 
numerous teachers to pubhsh separately, as a textbook in plane 
trigonometry, the material on trigonometry and logarithms of 
the text mentioned. 

The present textbook is the direct outcome of these requests. 
Of course, such separate publication of material taken out of 
the body of another book necessitated some changes and an in- 
troductory chapter. As a matter of fact, however, we have 
found it desirable to make a number of changes and additions 
not required by the necessities of separate publication. As a 
result fully half of the material has been entirely rewritten, with 
the purpose of bringing the text abreast of the most recent 
tendencies in the teaching of trigonometry. 

There is an increasing demand for a brief text emphasizing the 
numerical aspect of trigonometry and gi\Hing only so much of the 
theory as is necessary for a thorough understanding of the 
numerical applications. The material has .■ therefore been ar- 
ranged in such a way that the first six chapters give the essen- 
tials of a course in numerical trigonometry and logarithmic 
computation. The remainder of the theory usually given in 
the longer courses is contained in the last two chapters. 

More emphasis than hitherto has been placed on the use of 
tables. For this purpose a table of squares and square roots 
has been added. Recent experience has emphasized the appli- 
cations of trigonometry in navigation. ^Ye have accordingly 
added some material in the text on navigation, have introduced 

V 



vi PREFACE 

the haversine, and have added a four-place table of haversines 
for the benefit of those teachers who feel that the use of the 
haversine in the solution of triangles is desirable. This material 
can, however, be readily omitted by any teacher who prefers 
to do so. 

J. W. Young, 
F. M. Morgan. 

Hanover, N.H., 
August, 1919. 



CONTENTS 



CHAPITER 

I. Introductory Conceptions 



II. The Right Triangle 

III. Simple Trigonometric Relations 

IV. Oblique Triangles . 
V. Logarithms .... 

VI. Logarithmic Computation 

VII. Trigonometric Relations 

VIII. Trigonometric Relations (continued) 

Tables 

Index ........ 



PAGES 

1-10 
11-31 
32-39 
40-49 
50-60 
61-74 
75-87 
88-103 

106-119 
121-122 



vu 



PLAKE TRIGO]N^OMETRY AND 
JN^UMERICAL COMPUTATION 

CHAPTEK I 
INTRODUCTORY CONCEPTIONS 

1. The Uses of Trigonometry. The word " trigonometry '^ 
is derived from two Greek words meaning " the measurement 
of triangles.'' A triangle has six so-called elements (or parts) ; 
viz., its three sides and its three angles. We know from our 
study of geometry that, in general, if three elements of a tri- 
angle (not all angles) are given, the triangle is completely 
determined.* Hence, if three such determining elements of a 
triangle are given, it should be possible to compute the remain- 
ing elements. The methods by which this can be done, i.e. 
methods for " solving a triangle,'' constitute one of the prin- 
cipal objects of the study of trigonometry. 

If two of the angles of a triangle are given, the third angle 
can be found from the relation A -\- B -{- C = 180° (A, B, and 
C representing the angles of the triangle) ; also, in a right tri- 
angle, if two of the sides are kno^n, the third side can be 
found from the relation a^ -\- h- = C' (a, b being the legs and c 
the hypotenuse). But this is nearly the limit to which the 
methods of elementary geometry will allow us to go in the 
solution of a triangle. 

Trigonometry! is the foundation of the art of surveying 

* What exceptions are there to this statement ? 

t Throughout this book we shaU confine ourselves to the subject of ** plane 
trigonometry," which deals with rectilinear triangles in a plane. " Spherical 
trigonometry" deals with similar problems regarding triangles on a sphere 
whose sides are arcs of great circles. 

B 1 



PLANE TRIGOXO:\IETRY 



[I, § 1 



and of much of the art of navigation. It is, moreover, of 
primary importance in practically every branch of pure and 
applied mathematics. Many of the more elementary applica- 
tions will be presented in later portions of this text. 

2. The << Shadow Method." The ancient Greeks employed 
the theory of similar triangles in the solution of a special type 
of triangle problem which it is worth our while to examine 
briefly, because it contains the germ of the theory of trigo- 
nometry. 

It is desired to find the height CA of a vertical tower stand- 
ing on a level plain. It is observed 
that at a certain time the tower casts a 
shadow 42 ft. long. At the same time 
a pole CA', 10 ft. long, held vertically 
with one end on the ground casts a 
shadow 7 ft. long. From these data 
the height of the tower is readily com- 
puted as follows : The right triangles 
ABC 2ind A'B'C are similar since AB 
= Z B\ (\Yhy ? ) Therefore we have 
CA CA' 10 




A 



or 











BC 


B'C 










CA 


CA' 
B'C 


The tower is 


then 60 ft. 


high. 


3. 


A 


«« Function " 


of an 


Angle 



^(7 = ^X42 = 60. 



From the point of view of 
our future study the important thing to notice in the solution 

CA CA' 

of the preceding article is the fact that the ratios , 

^ ^ BC B'C 

are equal, i.e. that the ratio of the side opposite the angle B to 

the side adjacent to the angle is determined by the size of the angle, 

and does not depend at all on any of the other elements of the 

triangle, jDrovided only it is a right triangle. 



I, § 3] INTRODUCTORY CONCEPTIONS 3 

Definition. Whenever a quantity depends for its value on 
a second quantity, the first is called a function of the second. 

Thus in our example the ratio of the side opi)osite an angle 
of a right triangle to the side adjacent is a quantity which 
depends for its value only on the angle ; it is, therefore, called' 
a function of the angle. This ratio is merely one of several 
functions of an angle which we shall define in the next 
chapter. By means of these functions the fundamental prob- 
lem of trigonometry can be readily solved. 

The particular function which we have discussed is called 
the tangent of the angle. Explicitly defined for an acute angle 
of a right triangle, we have 

. 4. p 1 side opposite the ans^le 

tangent of angle = ^ ^ r^ — ^—~• 

side adjacent to the angle 

If the angle £ in the preceding example were measured it 
would be found to contain oo°. In any right triangle then 
containing an angle of Bd"" we should find this ratio to be equal 
to y^-, or 1.43. If the angle is changed, this ratio is changed, 
but it is fixed for any given angle. If the angle is 45°, the 
tangent is equal to 1, since in that case the triangle is 
isosceles. 

The word tangent is abbreviated " tan/'* Thus we have 
already found tan 5o° = 1.43 and tan 45° = 1.00. Similarly 
to every other acute angle corresponds a definite number, 
which is the tangent of that angle. The values of the tan- 
gents of angles have been tabulated. We shall have occasion 
to use such tables extensively in the future. 

If a, h, c are the sides of a right triangle ABC with right 
angle at C and with the usual notation whereby the side a is 
opposite the angle A and side b opposite the angle B, the defi- 
nition of the tangent gives 

tan^ = -. 
a 



PLANE TRIGONOMETRY 



[I, §3 



From this we get at once, 



b = a tan B and a = - 



tan B 



These are our first trigonometric formulas. By means of 
them and a table of tangents we can compute either leg of a 
right triangle, if the other leg and an acute angle are given. 



EXERCISES 

1. What is meant by '' the elements of a triangle " ? by *' solving a 
triangle" ? 

2. A tree casts a shadow 20 ft. long, when a vertical yardstick with 
one end on the ground casts a shadow of 2 ft. How high is the tree ? 

3. A chimney is known to be 90 ft. high. How long is its shadow 
when a 9-foot pole held vertically with one end on the ground casts a 
shadow 5 ft. long ? 

4. Give examples from your own experience of quantities which are 
functions of other quantities. 

5. Define the tangent of an acute angle of a right triangle. Why does 
its value depend only on the size of the angle ? 

6. In the adjacent figure think of the line BA as rotating about the 
point B in the direction of the arrow, starting from 
the position j5C (when the angle B is 0) and assum- 
ing successively the positions BAi, BA2, BA^, .... 

Show that the tangent of the angle B is veiy 
small when B is veiy small, that tan B increases as 
the angle increases, that tan B is less than 1 as 
long as 5 is less than 45°, that tan 45° = 1, that 
tan B is greater than 1 if the angle is greater than 
45°, and that tan B increases without limit as B ap- 
proaches 90°. 

7. The following table gives the values of the tan- 
gent for certain values of the angle : 




angle 


10° 


20° 


30° 


40° 


50° 


60° 


70° 


tangent 


0.176 


0.364 


0.577 


0.839 


1.19 


1.73 


2.75 



I, § 4] INTRODUCTORY CONCEPTIONS 5 

By means of this table tind the other leg of a right triangle ABC from 
the elements given : 

(a) B = 50^ a = 10 (d) B = 20°, b = IS (g) B = 60°, a = 37 
(6) B = 70°, a = 16 (e) A = 30°, 6=5 (h) A z= 20°, a = 22 
(c) B = 40°, 6 = 24 (/) ^ = 10°, b = Q2 

8. From the data and the results of the preceding exercise find the 
other acute angle and the hypotenuse of each of the right triangles. 

4. Coordinates in a Plane. The student should already be 
familiar from his study of algebra with the method of locating 
points in a plane by means of coordinates. Since we shall 
often have occasion to use such a method in the future, we will 
recall it briefly at this point. 

The method consists in referring the points in question to 
two straight lines X'X and Y' T, at right angles to each other, 
which are called the axes of 
coordinates. X'X is usually 
drawn horizontally and is 
called the x-axis ; Y^ Y, which 
is then vertical, is called the 
y-axis. 

The position of any point 
P is completely determined 
if its distance (measured in 
terms of some convenient ^^^* ^ 

unit) and its direction from each of the axes is known. Thus 
the position of P^ (Fig. 2) is known, if we know that it is 4 
units to the right of the i/-axis and 2 units above the a;-axis. If 
we agree to consider distance measured to the right or upwards 
as positive, and therefore distance measured to the left or down- 
ward as negative ; and if, furthermore, we represent distances 
and directions measured parallel to the x-axis by x, and distances 
and directions measured parallel to the ?/-axis by y, then the 
position of Pi may be completely given by the specifications 
x = -|-4, y =-\- 2\ or more briefly still by the symbol (4, 2). 



Second Quadras 


It 


< 

First Quadrant 


Ms. 




f 1 


X> M^ 


1 M, X 


n 


Third Quadra 


nt 


yt^ourfh Quadrant 



6 PLANE TRIGONOMETRY [I, § 4 

Similarly, the point P2 in Fig. 2 is completely determined 
by the symbol (-3, 5). Observe that in such a symbol the x of 
the point is written first, the y second. The two numbers x 
and y, determining the position of a point, are called the 
coordinates of the point, the x being called the x- coordinate 
or abscissa, the y being called the y-cobrdinate or ordinate 
of the point. What are the coordinates of P3 and P^ in 
Eig. 2? 

The two axes of coordinates divide the plane into four regions 
called quadrants, numbered as in Fig. 2. The quadrant in 
which a point lies is completely determined by the signs of its 
coordinates. Thus points in the first quadrant are character- 
ized by coordinates ( + , +), those in the second by ( — , +), 
those in the third by ( — , — ), and those in the fourth by (+, — ). 

Square-ruled paper (so-called coordinate or cross section 
paper) is used to advantage in " plotting '' {i.e. locating) points 
by means of their coordinates. 

5. Magnitude and Directed Quantities. In the last article 
we introduced the use of positive and negative numbers, i.e. 
the so-called signed numbers, while in the preceding articles, 
where we were concerned with the sides and angles of triangles, 
we dealt only with unsigned numbers. The latter represent 
magnitude or size only (as a length of 20 ft.), while the former 
represent both a magnitude and one of two opposite direc- 
tions or senses (as a distance of 20 ft. to the left of a given 
line). We are thus led to consider two kmds of quantities • 
(1) magnitudes, and (2) directed quantities. Examples of the 
former are : the length of the side of a triangle, the weight of 
a barrel of flour, the duration of a period of time, etc. Ex- 
amples of the latter are : the coordinates of a point, the tem- 
perature (a certain number of degrees above or below zero), 
the time at which a certain event occurred (a certain numbev 
of hours before or after a given instant), etc. 



I, § 6] INTRODUCTORY CONCEPTIONS 7 

Geometrically, the (listinction between directed quantities 
and mere magnitudes corresponds to the fact that, on the one 
hand, we may think of the line segment AB as drawn from A to 
B or from jB to ^ ; and, on the other hand, we 
may choose to consider only the length of ' * *~~^ ' ^ 
such a segment, irrespective of its direction. ' 
Figure 3 exhibits the geometric representation 
of 5, + 5, and — 5. A segment whose direc- 
tion is definitely taken account of is called 3i~directed segment. 
The magnitude of a directed quantity is called its absolute 
value. Thus the absolute value of — 5 (and also of + 5) is 5. 
Observe that the segments OM^, M^P^ (Fig. 2) representing 
the coordinates of P^ are directed segments. 

6. Directed and General Angles. In elementary geometry 

an angle is usually defined as the figure formed by two half- 
lines issuing from a point. However, it is often more serviceable 
to think of an angle as being generated 
by the rotation in a plane of a half-line 
/ \ OP about the point as a pivot, start- 

,-'''"" \ ing from the initial position OA and 

^ ^ ending at the terminal position OB (Fig. 

4). We then say that the line OP has 
generated the angle AOB, Similarly, if OP rotates from the 
initial position OB, to the terminal position OA, then the angle 
BOA is said to be generated. Considerations similar to those 
regarding directed line segments (§ 5) lead us to regard one of 
the above directions of rotation as positive and the other as 
negative. It is of course quite immaterial which one of the 
two rotations we regard as positive, but 
w^e shall assume, from now on, that 
cotmterclockicise rotation is positive and 
clockwise rotation is negative. 

Still another extension of the notion Fig. 5 



8 



PLANE TRIGONOMETRY 



[I, §6 



of angle is desirable. In elementary geometry no angle greater 
than 360° is considered and seldom one greater than 180°. But 
from the definition of an angle just given, we see that the 
revolving line OP may make any number of complete revolu- 
tions before coming to rest, and thus the angle generated may 
be of any magnitude. Angles generated in this way abound 
in practice and are known as angles of rotation.^ 

When the rotation generating an angle is to be indicated, it is 
customary to mark the angle by means of an arrow starting at 
the initial line and ending at the terminal line. Unless some 
such device is used, confusion is liable to result. In Fig. 6 




1110 



Fig. 



angles of 30°, 390°, 750°, 1110°, are drawn. If the angles were 
not marked one might take them all to be angles of 30°. 

7. MeaSlirement of Angles. For the present, angles will be 
measured as in geometry, the degree (°) being the unit of measure. A 
complete revolution is 360°. The other units in this system are the 
minute ('), of which 60 make a degree, and the second ('^), of which 60 
make a minute. This system of units is of great antiquity, having been 
used by the Babylonians. The considerations of the previous article then 
make it clear that any real number, positive or negative, may represent an 
angle, the absolute value of the number representing the magnitude of 
the angle, the sign representing the direction of rotation. 



X 

p' 
Fig. 7 



^k; 



Consider the angle XOP = 6^ whose vertex coincides with the origin 
of a system of rectangular coordinates, and whose initial line OX coin- 

*For example, the minute hand of a clock describes an angle of —180° 
n 30 minutes, an angle of — 540° in 90 minutes, and an angle of — 720° in 120 
ninutes. 




I, § 8] INTRODUCTORY CONCEPTIONS 9 

cides with the positive half of the ic-axis (Fig. 7). The angle $ is then 
said to be in the first, second, third, or fourth quadrant, according as its 
terminal line OP is in the first, second, third, or fourth quadrant. 

8. Addition and Subtraction of Directed Angles. The 

meaning to be attached to the sum of two directed angles is analogous to 
that for the sum of two directed 

line segments. Let a and b be /^ /^ 

two half -lines issuing from the ^ ^^ 

same point O and let (a6) repre- 
sent an angle obtained by rotat- 
ing a half -line from the position 
a to the position 6. Then if we 

have two angles (a6) and (6c) with the same vertex O, the sum (a6) -|- (6c) 
of the angles is the angle represented by the rotation of a half -line from 
the position a to the position 6 and then rotating from the position 6 to the 
position c. But these two rotations are together equivalent to a single rota- 
tion from a to c, no matter what the relative positions of a, 6, c may have 
been. Hence, we have for any three half-Unes a, 6, c issuing from a point 0, 
(1) {ah)\{hc) = {^ac), (a6) + (ba)=: 0, (a6) = (c&)-(ca). 
It must be noted, however, that the equality sign here means " equal, 
except possibly for multiples of 260^.^^ The proof of the last relation is 
left as an exercise. 

EXERCISES 

1. On square-ruled paper draw two axes of reference and then plot the 
.following points: (2, 3), (- 4, 2), (- 7, - 1), (0, - 3), (2, - 5), (5, 0). 

2. What are the coordinates of the origin ? 

3. Where are all the points for which x = 2? x=— 3? ?/=— 1? 
2/ = 4? x = 0? 

4. Show that any point P on the ^/-axis has coordinates of the form 
(0, y). What is the form of the coordinates of any point on the x-axis ? 

5. A right triangle has the vertex of one acute angle at the origin and 
one leg along the x-axis. The vertex of the other acute angle is at 
(7, 10). What is the tangent of the angle at O ? 

6. What angle does the minute hand of a clock describe in 2 hours 
and 30 minutes ? in 4 hours and 20 minutes ? 

7. Suppose that the dial of a clock is transparent so that it may be 
read from both sides. Two persons stationed at opposite sides of the dial 
observe the motion of the minute hand. In what respect will the angles 
described by the minute hand as seen by the two persons differ? 



10 PLANE TRIGONOIviETRY [I, § 8 

8. In what quadrants are the following angles : ST"" ? 135^ ? — 325° ? 
540^? 1500°? -270°? 

9. In what quadrant is ^/2 if ^ is a positive angle less than 360° and in 
the second quadrant ? third quadrant ? fourth quadrant ? 

10. By means of a protractor construct 27° + 85° +(— 30°) -f 20° + 
(-45°). 

11. By means of a protractor construct — 130° + 56° — 24°. 



CHAPTER II 



THE RIGHT TRIANGLE 



9. Introduction. At the beginning of the preceding chap- 
ter we described the fundamental problem of trigonometry to 
be the " solution of the triangle/' i.e. the problem of com- 
puting the unknown elements of a triangle when three of the 
elements (not all angles) are given. This problem can be 
solved by finding relations between the sides and angles of a 
triangle by means of which it is possible to express the un- 
known elements in terms of the known elements. In order 
to establish such relations, it has been found desirable to 
define certain functions of an angle. One such function — the 
tangent — was introduced in § 3 by way of preliminary illus- 
tration. 

In the present chapter, we shall give a new definition of the 
tangent of an angle and also define two other equally impor- 
tant functions — the sine and the cosine. It should be noted 
that the definition given for the tangent in § 3 applies only to 
an acute angle of a right triangle. For the purposes of a sys- 
tematic study of trigonometry we require a more general defini- 
tion, which will apply to any angle, positive or negative, and 
of any magnitude. Such definitions are given in the next 
article, in which the notion of a system of coordinates plays a 
fundamental role, the notion of a triangle not being introduced 
at all. After considering some of the consequences of our 
definitions in §§ 11-13, we consider the way in which these 
definitions enable us to express relations between the sides 
and angles of a right triangle. These results are then imme- 
diately applied to the solution of numerical problems by means 
of tables and to applications in surveying and navigation. 

11 



12 



PLANE TRIGONOMETRY 



[11, § 10 



10. The Sine, Cosine, and Tangent of an Angle. We 

may now define three of the functions referred to in § 3. To 
this end let d = XOP (Fig. 9) be any directed angle, and let 




JcyL^ 



us establish a system of rectangular coordinates in the plane 
of the angle such that the initial side OX of the angle is the 
positive half of the a^axis, the vertex being at the origin and 
the 2/-axis being in the usual position with respect to the 
a;-axis. Let the units on the two axes be equal. Finally, let 
P be any point other than on the terminal side of the angle 
6, and let its coordinates be (a?, y). The directed segment 
OP = r is called the distance of P and is always chosen posi- 
tive. The coordinates x and y are positive or negative accord- 
ing to the conventions previously adopted. We then define 
ordinate of P y 



The sine of Q = 



The cosine of Q = 
The tangent of Q = 



distance of P " 
abscissa of P 
distance of P" 
ordinate of P 



X 

y 



-, . p T^=-j provided x^ 0.^ 

abscissa oi P x 

These functions are usually written in the abbreviated forms 
sin 6, cos 9, tan 6, respectively ; but they are read as " sine 0,'^ 
" cosine 6,'' " tangent ^." It is very important to notice that 
the values of these functions are independent of the position 
of the point P on the terminal line, For let P (x\ y') be any 
other point on this line. Then from the similar right triangles 
ocyrf and x^y'r' it follows that the ratio of any two sides 
of the triangle xyr is equal in magnitude and sign to the 

* Prove that x and y cannot be zero simultaneously. 

t Triangle xyz means the triangle whose sides are x, y^ z. 



II, § 11] 



THE RIGHT TRIANGLE 



13 



ratio of the corresponding sides of the triangle x'y'r'. There- 
fore the values of the functions just defined depend merely 
on the angle 6. They are one-valued functions of 6 and are 
called trigonometric functions. 

Since the values of these functions are defined as the ratios 
of two directed segments, they are abstract numbers. They 
may be either positive, negative, or zero. Remembering that r 
is always positive, we may readily verify that the signs of the 
three functions are given by the following table. 



Quadrant 
Sine . . 
Cosine . 
Tangent 



1 

+ 
+ 
+ 



2 

+ 



+ 



4 

+ 



11. Values of the Functions for 45°, 135°, 225°, 315°. In 

each of these cases the triangle ocyr is isosceles. AYhy? 
Since the trigonometric functions are independent of the 
position of the point P on the terminal line, we may choose 
the legs of the right triangle xyr to be of length unity, which 



t^-- 






Fig. 10 



gives the distance OP as V2. Figure 10 shows the four angles 
with all lengths and directions marked. Therefore, 



sin 4o = — -, 

V2 



sin 135° : 



V2' 



sin 225° = --^, 

V2 

sin 315° = -—, 

V2 



cos 45° = - 



V2' 



cos 315° = 



vr 



tan 45° = 1, 



cos 135° = --!-, tan 135° = - 1, 

V2 

cos 225° = - — , tan 225° = 1, 

V2' 



tan 315° = - 1. 



14 



PLANE TRIGONOMETRY 



[II, § 12 



12. Values of the Functions for 30°, 150°, 210°, 330°. Erom 
geometry we know that if one angle of a right triangle con- 
tains 30°, then the hypotenuse is double the shorter leg, 
which is opposite the 30° angle. Hence if we choose the 
shorter leg (ordinate) as 1, the hypotenuse (distance) is 2, 



Ml it<^^ 



>/7 -^ 






S30 



fSf' 



Fig. 11 



and the other leg (abscissa) is V3. Figure 11 shows angles of 
30°, 150°, 210°, 330° with all lengths and directions marked. 
Hence we have 



sin 30° = :", 

2' 



sin 150° = - 



sin 210° = --, 

2'- 



sin 330° = - ±, 

2' 



cos 30° = 
cos 150° = 
cos 210° = 
cos 330° = 



V3 

2 ' 

V3 

2 ' 

V3 



V3 



tan 30° = 



V3 



tan 150°=-- 



V3 



-^-^, tan 210° = - 

2 ' 



V3 



tan 330° = 



V3 



13. Values of the Functions for 60°, 120°, 240°, 300°. It is 

left as an exercise to construct these angles and to prove that 



sin 60° = ^, 

2 ' 

sin 120°=^, 



cos 60°=-, 

2' 



cos 120° = ■ 



"2' 



tan 60°=V3, 
tanl20° = -V3, 



sin 240° = - ^, cos 240° = - -, tan 240° = V3, 

2 2 



sin 300° = - 



V5 



cos 300°=-, 

2' 



tan 300° = - V3. 



II, § 14] 



THE RIGHT TRIANGLE 



15 



14. Sides and Angles of a Right Triangle. Evidently any 
right triangle ABC can be so placed in a system of coordi- 
nates that the vertex of either acute 
angle coincides with the origin 
and that the adjacent leg lies along 
the positive end OX of the ic-axis 
(Fig. 12). The follov/ing relations 
then follow at once from the defini- 
tions of the sine, cosine, and tangent 
of § 10. 

In any right trimigle, the trigonometric functions of either acute 
angle are given by the ratios : 




the sine • 



the cosine = 



side opposite the angle 
hypotenuse ' 

side adjacent to the angle 



the tangent = 



hypotenuse 
side opposite the angle 



side adjacent to the angle ' 

These relations are fundamental in all that follows. They 
should be firmly fixed in mind in such a way that they can be 
readily applied to any right triangle in what- 
ever position it may happen to be (for example 
as in Fig. 13). The student should be able to 
reproduce any of the following relations with- 
out hesitation whenever called for. They 
should not be memorized, but should be read 
from an actual or imagined figure : 




Fig. 13 



sin J. = - 



sin J5 = - , 
c 



cos A = -, 
c 



cos 5 = - 



tan ^ = - , 



Also the known relation : 



tan B = ~, 
a 



c2 = a2 + 62, 



16 



PLANE TRIGONOMETRY 



[II, § 14 



If any two elements (other than the right angle) of a right 
triangle are given, we can then find a relation connecting these 
two elements with any unknown element, from which relation 
the unknown element can be computed. 



15. Applications. The angle which a line from the eye to 
an object makes with a horizontal line in the same vertical 
plane is called an angle of elevation or an angle of depression, 




Fig. U 



Horizontal 




according as the object is above or below the eye of the ob- 
server (Fig. 14). Such angles occur in many examples. 

Example 1. A man wishing to know the distance between two points 
A and B on opposite sides of a pond locates a point C on the land (Fig. 
15) such that J.C = 200 rd., angle C = 30^ and angle B = 90°. Fmd the 
distance AB. 

AB _ 
AG 

AB = AC 'Sin G 
= 200 . sin 30° 
= 200 . 1 



Solution : 



- = sin G. 



(Why ?) 



100 rd. 




Fig. 15 



Example 2. Two men stationed at points A and G 800 yd. apart and 
in the same vertical plane with a balloon J5, observe simultaneously the 
angles of elevation of the balloon to be 30° and 45° respectively. Find the 
height of the balloon. 

Solution : Denote the height of the balloon DB by y, and let DG = x; 
then AD = 800 - x. 




Fig. 16 



II, § 15J THE RIGHT TRIANGLE 17 

Since tan 45^ = 1, we have 1 



X 

1 _ y 
and since tan 30^ = 1/^'^^ ^^'^ \vii\Q — ^^ _ * 

Therefore x = y and 800 — x = ?/ VS. 

'Solving these equations for y, we have y = = 292.8 yd. 

V3 + 1 

EXERCISES 

1. In what quadrants is the sine positive ? cosine negative ? tangent 
positive ? cosine positive? tangent negative ? sine negative ? 

2. In w^hat quadrant does an angle lie if 

"(a) its sine is positive and its cosine is negative ? 

(6) its tangent is negative and its cosine is positive? 

(c) its sine is negative and its cosine is positive ? 

(d) its cosine is positive and its tangent is positive ? 

3. T\Tiich of the following is the greater and wiiy : sin 49° or cos 49° ? 
sin 35° or cos 35° ? 

4. If ^ is situated between 0° and 360°, how many degrees are there in 
e if tan ^ = 1 ? Answer the similar question for sin ^ = i ; tan d = —\. 

5. Does sin 60° = 2 . sin 30° ? Does tan 60° = 2 • tan 30° ? What 
can you say about the truth of the equality sin 2 ^ = 2 sin ^ ? 

6. The Washington Monument is 555 ft. high. At a certain place in 
the plane of its base, the angle of elevation of the top is 60°. How far is 
that place from the foot and from the top of the tower ? 

7. A boy whose eyes are 5 ft. from the ground stands 200 ft. from a 
flagstaff. From his eyes, the angle of elevation of the top is 30°. How 
high is the flagstaff ? 

8. A tree 38 ft. high casts a shadow 38 ft. long. What is the angle 
of elevation of the top of the tree as seen from the end of the shadow ? 
How far is it from the end of the shadow to the top of the tree ? 

9. From the top of a tower 100 ft. high, the angle of depression of 
two stones, which are in a direction due east and in the plane of the base 
are 45° and 30° respectively. How far apart are the stones ? 

.4718. 100(V3 - 1)= 73.2 ft. 
c 



18 



PLANE TRIGONOMETRY 



[II, § 15 



10. Find the area of the isosceles triangle in which the equal sides 10 
inches in length include an angle of 120^. Ans. 25 V3 = 43.3 sq. in. 

11. Is the formula sin 2 ^ = 2 sin 6 cos $ true when 6 = 30° ? 60° ? 
120° ? 

12. From a figure prove that sin 117^ = cos 27°. 

13. Determine whether each of the following formulas is true when 
e = 30°, 60°, 150°, 210° : 



1 + tan2 e = 
1 + - ' 



1 



cos^ 6 ' 
1 



tan^(9 sin^^' 
sin2(9 -f cos- (9 = 1. 

14. Let Pi(Xi, ?/i) and P2(X2, ?/2) he any two points the distance be- 
tween which is r (the units on the axes being equal) . If ^ is the angle 
that the hne P1P2 makes with the x-axis, prove that 

^2 -Xi ^ y ^-Vi ^ 2 r. 
cos d sin d 



16. Computation of the Value 
Function from that of Another. 



of One Trigonometric 



J{1 



Fig. 17 



Example 1. • Given that sin ^ = |, find the 
values of the other functions. 

Since sin is positive, it follows that d is 
an angle in the first or in the second quad- 
rant. Moreover, since the value of the sine 
is I, then y = Z - k and r = 5 • ^*, where k is 
any positive constant different from zero. (Why ?) It is, of course, 
immaterial what positive value we assign to A*, so we shall assign the 
value 1. We know, however, that the abscissa, ordinate, and distance 
are connected by the relation x^ + 7/2 = r^, and hence it follows that 
X = ± 4. Figure 17 is then self-explanatory. Hence we have, for the first 
quadrant, sin ^ = |, cos ^ = |, and tan ^ = f ; for the second quadrant, 
sin ^ = f , cos ^ = — I, tan d = — I. 

Example 2. Given that sin^r=y\ and that tan ^ 
is negative, find the other trigonometric functions of 
the angle 6. 

Since sin 6 is positive and tan 6 is negative, 6 must 
be in the second quadrant. We can, therefore, con- 
struct the angle (Fig. 18), and we obtain sin ^ = y^j , 



cos ^ =— i|, tan e : 



_5_ 
12* 



Fig. 18 



II, § 17] 



THE RIGHT TRIANGLE 



19 



17. Computation for Any Angle. Tables. The values of 
the trigonometric functions of any angle may be computed by 
the graphic method. For 
example, let us find the 
trigonometric functions of 
35°. We tirst construct 
on square-ruled paper, 
by means of a protractor, 
an angle of 35° and choose 
a point P on the ter- 
minal line so that OP 
shall equal 100 units. 
Then from the figure we 
find that OM=z 82 units 
and MP = 57 units. 
Therefore 




30 iO 50 60 70 80 90 100 

Fig. 19 



sin 35° = i¥o = ^-^^^ cos 35° = -f-^-^ = 0.82, tan 35° = fj = 0.70. 

The tangent may be found more readily if we start by tak- 
ing OA = 100 units and then measure AB. In this case, 
AB = 70 units and hence tan 35^ = j^-^ = 0.70. 

It is at once evident that the graphic method, although 
simple, gives only an approximate result. However, the values 
of these functions have been computed accurately by methods 
beyond the scope of this book. The results have been put in 
tabular form and are known as tables of natural trigonometric 
functions. Such tables and how to use them will be discussed 
in the next article. 

Figure 20 makes it possible to read off the sine, cosine, or 
tangent of any angle between 0° and 90° with a fair degree of 
accuracy. The figure is self-explanatory. In reading off 
values of the tangent use the vertical line through 100 for angles 
up to 55°, and the line through 10 for angles greater than 55°. 
Its use is illustrated in some of the following exercises. 



20 



PLANE TRIGONOMETRY [II, § 17 




iO 20 30 iO 50 60 70 SO 90 100 

Fig. 20. — Graphical Table of Trigonometric Functions 



II, § 18] THE RIGHT TRIANGLE 21 

EXERCISES 

Find the other trigonometric functions of the angled when 

1. tan e = -S. 3. cos (9 = ^f 5. sin (9 = J. 

2. sin (9 = - |. 4. tan ^ = f . 6i cos ^ = — -J. 

7. sin 6 = ^ and cos d is negative. 

8. tan = 2 and sin 6 is negative. 

9. sin ^ = — i and tan 6 is positive. 

10. cos d = i and tan 6 is negative. 

11. Can 0.6 and 0.8 be the sine and cosine, respectively, of one and 
the same angle ? Can 0.5 and 0.9 ? Ans. Yes ; no. 

12. Is there an angle whose sine is 2 ? Explain. 

13. Determine graphically the functions of 20^, 38^ 70°, 110^. 

14. From Fig. 20, find values of the following : 

sin 10°, cos 50°, tan 40°, sin 80°, tan 70°, cos 32°, tan 14°, sin 14°. 

15. A tower stands on the shore of a river 200 ft. wide. The angle of 
elevation of the top of the tower from the point on the other shore exactly 
opposite to the tower is such that its sine is |. Find the height of the 
tower. 

16. From a ship's masthead 160 feet above the water the angle of de- 
pression of a boat is such that the tangent of this angle is ^^2* Find the 
distance from the boat to the ship. Ans. 640 yards. 

18. Use of Tables of Trigonometric Functions. Examina- 
tion of the tables of ^' Four Place Trigonometric Functions '' 
(p. 112) shows columns headed '^ Degrees/' " Sine/' " Tangent/' 
" Cosine/' and under each of the last three named a column 
headed " Value " (none of the other columns concern us at pres- 
ent). Two problems regarding the use of these tables now 
present themselves. 

1. To find the value of a function when the angle is given, 
(a) Find the value of sin 15° 20'. In the column headed 
" Degrees " locate the line corresponding to 15° 20' (p. 113) ; on 
the same line in the " value " column for the " Sine/' we read 
the result : sin 15^ 20' = 0.2644. On the same line, by using 
the proper column, we find tan 15° 20' = 0.2742, and cos 15° 20' 
= 0.9644. 



22 PLANE TRIGONOMETRY [II, § 18 

(b) Find the value of tan 57° 50'. The entries in the 
column marked " Degrees " at the top only go as far as 45° 
(p. 116). But the columns marked " Degrees '' at the bottom 
contain entries beginning with 46° (p. 116) and running back- 
wards to 90° (p. 112). In using these entries we must use the 
designations at the bottom of the columns. Thus on the line 
corresponding to 57° 50' (p. 115) we find the desired value : 
tan 57° 50' = 1.5900. Also sin 57° 50' = 0.8465, and cos 57° 
50' = 0.5324. 

(c) Find the value of sin 34° 13'. This value lies between 
the values of sin 34° 10' and 34° 20'. We find for the latter 

sin 34° 10' = 0.5616 

sin 34° 20' = 0.5640 

Difference for 10' = 0.0024 

Assuming that the change in the value of the function 
throughout this small interval is proportional to the change in 
the value of the angle, we conclude that the change for 1' in the 
angle would be 0.00024. For 3', the change in the value of the 
function would then be 0.00072. Neglecting the 2 in the last 
place (since we only use four places and the 2 is less than 5), 
we find sin 34° 13' = 0.5616 + 0.0007 = 0.5623. This process is 
called interpolation. With a little practice all the work in- 
volved can and should be done mentally ; Le. after locating the 
place in the table (and marking it with a finger), we observe 
that the " tabular difference '' is '^ 24 '' ; we calculate mentally 
that .3 of 24 is 7.2, and then add 7 to 5616 as we write down 
the desired value 0.5623. 

Similarly we find tan 34° 13' = 0.6800 (the correction to be 
added is in this case 12.9 which is " rounded off '' to 13) and 
cos 34° 13' = 0.8269. (Observe that in this case the correction 
must be subtracted. Why ?) 

2. To find the angle ivhen a value of a function is given. 
Here we proceed in the opposite direction. Given sin A = 



II, § IS] THE RIGHT TRIANGLE 23 

0.3289 ; find A. An examination of tlie sine column shows 
that the given value lies between sin 19° 10' (= 0.3283) and 
sin 19° 20'(= 0.3311). We note the tabular difference to be 28. 
The correction to be applied to 19^ 10' is then ^\ of 10' = f f ' 
= i^^' = 2.1'. Hence A = 19° 12.1'. (With a four place table 
do not carry your interpolation farther than the nearest tenth 
of a minute.) (See § 20.) 

EXERCISES 

1. For practice in the use of tables, verify the following : 

(a) sin 18° 20' = 0.3145 (d) sin 27M-1' =0.4576 (6^) sin 62° 24M = 0.886 2 
(6) cos 37'^ 30' =0.7934 (e) cos 34° 11' =0.8272 (/i) cos 59° 46' .2 =0.5034 
(c) tan 75° 50' = 3.9617 (/) tan 68° 21' = 2.5173 (i) tan 14° 55'.6 =0.2665 
Assume first that the angles are given and verify the values of the 
functions. Then assume the values of the functions to be given and 
verify the angles. 

2. A certain railroad rises 6 inches for every 10 feet of track. What 
angle does the track make with the horizontal ? 

3. On opposite shores of a lake are two flagstaff s A and B. Per- 
pendicular to the line AB and along one shore, a line BC = 1200 ft. is 
measiu-ed. The angle ACB is observed to be 40° 20'. Find the distance 
between the two flagstaffs. 

4. The angle of ascent of a road is 8°. If a man walks a mile up the 
road, how many feet has he risen ? 

5. How far from the foot of a tower 150 feet high must an observer, 
6 ft. high, stand so that the angle of elevation of its top may be 23°. 5 ? 

6. From the top of a tower the angle of depression of a stone in the 
plane of the base is 40° 20' . What is the angle of depression of the stone 
from a point halfway down the tower? 

7. The altitude of an isosceles triangle is 24 feet and each of the equal 
angles contains 40° 20'. Find the lengths of the sides and area of the 
triangle. 

8. A flagstaff 21 feet high stands on the top of a cliff. From a point 
on the level with the base of the cliff, the angles of elevation of the top 
and bottom of the flagstaff are observed. Denoting these angles by u 
and /3 respectively, find the height of the cUff in case sin a = -/y and 
cos/3 = ^f. Ans, 75 feet. 



24 PLANE TRIGONOMETRY [II, § 18 

9. A man wishes to find the height of a tower CB which stands on a 
horizontal plane. From a point A on this plane he finds the angle of ele- 
•vation of the top to be such that sin CAB = |. From a point A' which 
is on the line AG and 100 feet nearer the tower, he finds the angle of 
elevation of the top to be such that tan CA'B = J. Find the height of 
tiie tower. 

10. Find the radius of the inscribed and circumscribed circle of a regu- 
lar pentagon whose side is 14 feet. 

11. If a chord of a circle is two thirds of the radius, how large an 
angle at the center does the chord subtend ? 

19. Computation with Approximate Data. Significant 
Figures. The numerical applications of trigonometry (in sur- 
veying, navigation, engineering, etc.) are concerned with com- 
puting the values of certain unknown quantities (distances, 
angles, etc.) from known data which are secured by measure- 
ment. Now, any direct measurement is necessarily an approxi- 
mation. A measurement may be made with greater or less 
accuracy according to the needs of the problem in hand — but 
it can never be absolutely exact. Thus, the information on a 
signpost that a certain village is 6 miles distant merely 
means that the distance is 6 miles to the nearest mile — i.e. that 
the distance is between 5^ and 6^ miles. Measurements in a 
physical or engineering laboratory need sometimes to be made to 
the nearest one thousandth of an inch. For example the bore 
of an engine cylinder may be measured to be 3.496 in., which 
means that the bore is between 3.4955 in. and 3.4965 in. 

A simple convention makes it possible to recognize at a 
glance the degree of accuracy implied by a number represent- 
ing an approximate measure (either direct or computed). This 
convention consists simply in the agreement to write no more 
figures than the accuracy warrants. Thus in arithmetic 6 and 
6.0 and 6.00 all mean the same thing. This is not so, when 
these numbers are used to express the result of measurement 
or the result of computation from approximate data. Thus 6 
means that the result is accurate to the nearest unit, 6.0 that 



II, § 20] THE RIGHT TRIANGLE 25 

it is accurate to the nearest tenth of a unit, 6.00 to the nearest 
hundredth of a unit. 

These considerations have an important bearing on practical 
computation. If the side of a square is measured and found 
to be 3.6 in. and the length of the diagonal is cbmputed by 
the formula : diagonal = side x V2, it would be wrong to write 
= 3.6 X V2 = 3.6 X 1.4142 = 5.09112 in. The correct result 
is 5.1 in. For the computed value of the diagonal cannot be 
more accurate than the measured value of the side. The result 
5.09112 must therefore be " rounded off ^' to two significant 
figures, which gives 5.1. As a matter of fact for the purpose 
of this problem V2 = 1.4142 should be rounded before multi- 
plication to \/2 = 1.4 ; thereby reducing the amount of labor 
necessary. 

A number is '' rounded off,*' by dropping one or more digits 
at the right and, if the last digit dropped is 5^, 6, 7, 8, or 9 
increasing the preceding digit by 1.^ Thus the successive 
approximations to tt obtained by rounding of 3.14159 ••• are 
3.1416, 3.142, 3.14, 3.1, 3. 

20. The Number of Significant Figures of a number (in the 
decimal notation) may now be defined as the total number of 
digits in the number, except that if the number has no digits 
to the right of the decimal point, any zeros occurring between 
the decimal point and the first digit different from zero are 
not counted as significant. Thus, 34.06 and 3,406,000 are both 
numbers of four significant figures : while 3,406,000.0 is a 
number of eight significant figures.! 

* In roundino: off a 5 computers round off to an even digit. Thus .1.415 
would be rounded to 1.42, whereas 1.445 would be rounded to 1.44. If this 
rule is used consistently the errors made will tend to compensate each other. 

t Confusion will arise in only one case. For example, if 3999.7 were 
rounded by droppinoj the 7 we should write it as 4000 which according to the 
above definition would have only 1 significant figure, whereas we know from 
the way it was obtained that all four figures are significant. In such a case 
we may underscore the zeros to indicate they are significant or use some 
other device. 



26 PLANE TRIGONOMETRY [II, § 20 

In any computation involving multiplication or division the 
number of significant figures is generally used as a measure of 
the accuracy of the data. A computed result should not in 
general contain more significant figures than the least accurate 
of the data. But computers generally retain one additional 
figure during the computation and then properly round off the 
final result. Even then the last digit may be inaccurate — but 
that is unavoidable. 

The following general rules will be of use in determining 
the degree of accuracy to be expected and in avoiding useless 
labor : 

1. Distances expressed to two significant figures call for 
angles expressed to the nearest 30' and vice versa. 

2. Distances expressed to three significant figures call for 
angles expressed to the nearest 5', and vice versa, 

3. Distances expressed to four significant figures call for 
angles expressed to the nearest minute, and vice versa. 

4. Distances expressed to five significant figures call for 
angles expressed to the nearest tenth of a minute, and vice 
versa. 

In working numerical problems the student should use every 
safeguard to avoid errors. Neatness and systematic arrange- 
ment of the work are important in this connection. All work 
should be checked in one or more of the following ways. 
1. Gross errors may be detected by habitually asking oneself : 
Is this result reasonable or sensible ? 2. A figure drawn to 
scale makes it possible to measure the unknown parts and to 
compare the results of such measurements with the computed 
results. 3. An accurate check can often be secured with com- 
paratively little additional labor by computing one of the 
quantities from two different formulas or by verifying a 
known relation. For example, if the legs a, 6 of a right tri- 
angle have been computed by the formulas a = c sin ^ and 
6 = c cos A, we may check by verifying the relation a^ + 6^ = c^. 



n, § 21] 



THE RIGHT TRIANGLE 



27 




Example. A stniii^ht road is to be built from a point A to a point Ji 

which is 5.02 miles east and 8.27 miles north of 

A. What will be the direction of the road and 

its length ? 

T7 1 * I 6.92 , „ 8.27 

Formulas : tan A = ; AB = 

8 27 cos A 

Therefore tan A = 0.716 and A = 35^ 35', 

cos A = 0.813 AB = 10.17.* 

Check by a2 + 62 = ^2. 

From a table of squares (p. 107, see § 21) 

(5.92)2= 35.05 

(8.27)2 = 68.39 (10.17)2 = 103.4. 

103.4 

21. Use of Table of Squares. Square Roots. The table 
of squares of numbers (p. 106) may be used to facilitate com- 
putation. In the example of the last article, we required the 
square of 5.92. We find 5.9 on p. 107 in the left-hand column 
and find the third digit 2 at the head of a certain column. At 
the intersection of the line and column thus determined we 
find the desired result (5.92)2 ^ 35,95, The square of 8.27 is 
found similarly at the intersection of the line corresponding 
to 8.2 and the column headed 7. To find (10.17)^, we find the 
line corresponding to 1.0 (the first two digits, neglecting the 
decimal point) and find (1.01)^ = 1.020 and (1.02)2 = 1.040. 
By interpolating, as explained in § 18, we find (1.017)2 = 1.034. 
Now shifting the decimal point one place in the "number" 
requires a corresponding shift of two places in the square. 
Hence, (10.17)2 = 103.4. 

The table can also be used to find the square root of a num- 
ber. Thus to find V2 we find, on working backwards in this 
table, that 2 lies between 1.988 [=(1.41)^] and 2.016 [ = (1.42)2]. 
By interpolation we then find V2 = 1.414, correct to four 
significant places. [Tabular difference = 28 ; correction = -^^ 
= 4 in the fourth place.] 



♦The retention of four significant figures in AB is justified because the 
number is so small at the left. 



28 PLANE TRIGONOMETRY [II, § 21 



EXERCISES 

1. From an observing station 357 ft. above the water, the angle of 
depression of a ship is 2° 15^ Elnd the horizontal distance to the ship in 
yards. 

2. A projectile falls in a straight line making an angle of 25° with the 
horizontal. AYill it strike the top of a tree 24 meters high which is 72 meters 
from the point where the projectile would strike the ground ? 

3. At a point 372 ft. from the foot of a cliff surmounted by an observa- 
tion tower the angle of elevation of the top of the tower is 51° 25', and of 
the foot of the tower 31° 55'. Find the height of the cliff and of the 
tower. 

4. How far from the foot of a flagpole 130 ft. high must an observer 
stand so that the angle of elevation of the top of the pole will be 25° ? 

5. GA is a horizontal line, T is a point vertically above A;B?i point 

A C 

vertically below A. The angle BGA in minutes is Find Z BGT 

4000 

in degrees and minutes, given GA = 10,340 meters , AT = 416.4 meters. 

6. It is desired to find the height of a wireless tower situated on the 
top of a hill. The angle subtended by the tower at a point 250 ft. below 
the base of the tower and at a distance measured horizontally of 2830 ft. 
from it is found to be 2° 42'. Find the height of the tower. 

7. From a tower 428.3 ft. high the angles of depression of two objects 
situated in the same horizontal line with the base of the tower and on the 
same side are 30° 22' and 47° 37'. Find the distance between them. 

8. The summit of a mountain known to be 13.260 ft. high is seen at 
an angle of elevation of 27° 12' from a camp located at an altitude of 
6359 ft. Compute the air-hne distance from the camp to the summit of 
the mountain. 

9. Two towns A and B^ of which B is 25 miles northeast of J., are to 
be connected by a new road. 11 miles of the road is constructed from 
A in the direction N. 21° E. ; what must be length and direction of the 
remainder of the road, assuming it to be straight ? 

22. Applications in Navigation. We shall confine ourselves 
to problems in plane sailing ; i.e. we shall assume that the dis- 
tances considered are sufficiently small so that the curvature of 
the earth may be neglected. 



11, § 22] 



THE RIGHT TRIANGL^] 



29 




Fig. 21 



Defixitiox. The course of a 

ship is the direction in which she 

is sailing. It is given either by 

the points of a mariner's compass 

(Fig. 21) as X. E. hy X. or in 

degrees and minutes measured 

dockicise from the north. Observe 

that a " point '^ on a mariner's 

compass is 11° 15'. Hence for 

example, the course of a ship 

could be given either as X. E. by 

N. or as 33° 45'. A course S. E. by S. is the same as a course 

of 146° 15'. 

The distance a ship travels on a given course is always given 
Departure ^ uautlcal mllcs or knots. A knot is the length 
of a minute of arc on the earth's equator. (The 
earth's circumference is then 360 x 60 = 21,600 
knots.) The horizontal component of the dis- 
tance is called the departure, the vertical com- 
ponent is called the difference in latitude. The 
departure is usually given in miles (knots), the 
difference in latitude in decrees and minutes. 




Fig. 22 



Example. A ship starts from a position in 22° 12' X. lati- 
tude, and sails 321 knots on a course of 31° 15'. Find the 
difference in latitude, the departure, and the latitude of the 
new position of the ship, 
diff. in lat. = distance times cosine of course 

= 321 cos 31° 15' 

= 321 X 0.855 = 274' = 4° 34'. 
departure = distance times sine of course 

= 321 sin 31° 15' 

= 321 X 0.519 = 167 knots. 




Since the ship is sailing on a course wliich increases the lati- 



30 PLANE TRIGONOMETRY [II, § 22 

tude, the latitude of the new position is 22° 12' + 4° 34' = 26° 
46' N. 

Knowing the difference in latitude and the departure, we are 
able to calculate the new position of the ship, if the original 
position is known. In the preceding example, we found the 
latitude of the new position from the difference in latitude. 
To find the difference in longitude from the departure is not 
quite so simple. As the latitude increases, a given departure 
implies an increasing difference in longitude. Only on the 
equator is the departure of one nautical mile equivalent to a 
difference in longitude of one minute. 

The adjacent figure shows a departure AB in latitude <f}. 
The difference in longitude (in minutes) corresponding to AB 
is clearly the number of nautical miles in 
CD, Xow arcs AB and CD are proportional 
to their radii FA and OC. Or, 

CD = ^ ^ AB=-^^. (Why ?) 
FA cos(/> ^ -^ ^ 

In practice, it is customary to take for <^ 
in the determination of difference in longi- 
tude the so-called middle latitude^ i.e. the 
latitude halfway between the original latitude and the final 
latitude. 

Thus in the preceding example, the original latitude was 
22° 12' N, the final latitude was 26° 46' N. The middle lati- 
tude is therefore i (22° 12' + 26° 46')= 24° 29'. Hence 

j.pp . T ., -, departure 

difference m longitude = -, , ^ . ^ ^. — = — -. — ^ 

cosine 01 middle latitude 

167 16^ = 184' = 3° 4'. 




cos 24° 29' ■" 0.910 



The determination of the position of a ship from its course 
and distance is known as dead reckoning. It is subject to con- 
siderable inaccuracy and must often in practice be checked by 



II, § 22] THE RIGHT TRIANGLE 31 

direct determination of position by observations on the sun 
or stars. 

EXERCISES 

1. A ship sails X. E. by E. at the rate of 12 knots per hour. Find the 
rate at which it is moving north. 

2. A ship sails N. E. by N. a distance of 578 miles. Find its departure 
and difference in latitude. 

3. A ship sails on a course of 73^ until its departure is 315 miles. Find 
the actual distance sailed. Find also its difference in latitude. 

4. A ship sails from latitude 47^ 15' N. 670 miles on a course N. W. 
by N. Find the latitude arrived at. 

6. A ship sails from latitude 30° 24' N. and after 25 hours reaches lati- 
tude 35° 26' X. Its course was X. X. W. Find the average speed of the 
ship. 

6. A vessel sails from lat. 24^ 30' X., long. 30^ 15 W., a distance of 692 
miles on a coiu-se of 32^20'. Find the latitude and longitude of its new 
position. 

7. A vessel sails from lat. 10^30' S., long. 167^ 20' W.. a distance of 
692 miles on a course of 152^ 30'. Find the latitude and longitude of its 
new position. 





ordinate of P 


y 


secant 6 = 


distance of F 
abscissa of P 


_ r 

X 


f»nt,a n P'pn t fi - 


abscissa of P 


X 




ordinate of P 


y 



CHAPTEE III 

SIMPLE TRIGONOMETRIC RELATIONS 

23. Other Trigonometric Functions. The reciprocals of 

the sine, the cosine, and the tangent of any angle are called, 
respectively, the cosecant, the secant, and the cotangent of 
that angle. Thus, 

^^ distance of P r . •-, -, , r.. 
--------^ /] (provided y =^ 0). 

(provided x ^ 0). 

(provided y ^0), 

These functions are written esc 6, sec 6, ctn 6. From the 
definitions follow directly the relations 

CSC 6 = , sec 6 = , ctn 9 = ; 

sin 6 cos 6 tan 6 

or 

CSC 6 * sin ^ = 1, sec d • cos 6 = 1, ctn 6 • tan 6 = 1. 

To the above functions may be added versed sine (written versin), the 
coversed sine (written coversin), and the external secant (written exsec), 
which are defined by the equations versin d = 1 — cos d^ coversin 6 = 
1 — sin ^, and exsec 6 = sec 6 — 1. Of importance in navigation and service- 
able in other applications (see § 38) is the haversine (written hav) 
which is defined to be equal to one half the versed sine ; i.e. 

have = 1(1 — COS0). 

24. The Representation of the Functions by Lines. Con- 
sider an angle 6 in each quadrant and about the origin draw 

32 



Ill, § 24] SIMPLE TRIGONOxMETRIC RELATIONS 33 




Fig. 24 



a circle of unit radius. Let P{x, y) be the point where the 
circle meets the terminal side of 6. Then 



sin (9 = ^=?/, cos (9 = 1 = a;, 



i.e. the sine is represented by the ordinate of P and the cosine 
by the abscissa. Hence the sine and cosine have respectively 
the same signs as the ordinate and abscissa of P. 

If we draw a tangent to the circle at the point A where the 




Fig. 25 



circle meets the a;-axis and let the terminal line of meet this 
tangent in Q, we have 

tan^ = ^=^Q, sec^ = ^=OQ. 

Note that when = 90°, 270°, and in general 90 + n . 360°, 
-270° + n . 360°, where n is any integer, there is no length AQ 
cut off on the tangent line and hence these angles have no 
tangents. 

If we draw a line tangent to the circle at the point B where 



34 



PLANE TRIGONOMETRY 



[III, § 24 



the circle cuts the y-a,xis and let the terminal line of cut 
this tangent in B, we have 

ctn ^ = ^ = BE, and esc (9 = — = OR. 




Fig. 26 



H Cotan. ^ 




1^ 


5^ 


s. 


/v 




\ 


/ *'V 




\ 


^(^ 


0^ 




M 


I 


/ 



EXERCISES 

1. From Fig. 24 prove sin^ 8 + cos^ ^ = 1. 

2. From Fig. 25 prove 1 + tan2 6 = sec^ ^. 

3. From Fig. 26 prove 1 + ctn2 6 = csc2 e, 

25. Relations among the Trigonometric Functions. As 

one might imagine, the six trigonometric functions sine, cosine, 
tangent, cosecant, secant, cotangent are connected by certain 
relations. We shall now find some of these relations. 

From Eig. 9 (§ 10) it is seen that for all cases we have 
(1) x^ + y^ = r\ 

If we divide both sides of (1) by r^, we have 

- + ^ = 1 (by hypothesis r=^0); 
or 



sin2 e + cos2 e = 1. 

Dividing both sides of (1) by x^, we have 



7/2 A.2 

1 + -. = ^ (if o; =5^0). 



Therefore, 



x^ x"- 
1 + tan2 e = sec2 6. 

Similarly dividing both sides of (1) by ]p- gives 



;+i=^ 



or 



ctn2 e + 1 = csc2 e 



(if2/=^0); 



Ill, § 26] SIMPLE TRIGONOMETRIC RELATIONS 35 
Moreover, we have 

y 

A V ^ sin 9 

tan e = ^ = - = ^, 

X X cos 6 

r 

and, similarly, 

' cos 6 
ctn9 = -T-5-. 
, sin 6 

26. Identities. By means of the relations just proved 
any expression containing trigonometric functions may be 
put into a number of different forms. It is often of the 
greatest importance to notice that two expressions, although 
of a different form, are nevertheless identical in value. (How 
was an "^ identity'' defined in algebra ?) 

The truth of an identity is usually established by reducing 
both sides, either to the same expression, or to two expres- 
sions which we know to be identical. The following examples 
will illustrate the methods used. 

Example 1. Prove the relation sec2 d + csc^ 6 = sec2 d csc2 d. 
We may write the given equation in the form 

^ + -^^ = sec2 ^ csc2 (9, 



cos2 6 sin2 e 
sin2 d + cos2 e 



~ sec2 d csc2 ^, 



which reduces to 



cos2 e sin2 e 

= = sec2 d csc2 d. 

cos2 e sin2 d 

sec2 d csc2 (9 = sec2 ^ csc2 d. 



Since this is an identity, it follows, by retracing the steps, that the 
given equality is identically true. 

Both members of the given equality are undefined for the angles 0^, 90°, 
180°, 270°, 360°, or any multiples of these angles. 

PQc2 

Example 2. Prove the identity 1 + sin ^ = 

^ 1 - sin ^ 

Since cos2 ^ = l — sin2 d, we may write the given equation in the form 

1 + sin ^ = ^ ~~ '^"^'^ ^ or 1 4- sin ^ = 1 + sin d. 
1 - sin ^ 



36 PLANE TRIGONOMETRY [III, § 26 

As in Example 1, this shows that the given equahty is identically true. 

The right-hand member has no meaning when sin ^ = 1 , while the left- 
hand member is defined for all angles. We have, therefore,- proved that 
the two members are equal except for the angle 90*^ or (4 m-|-1)90°, where 
n is any integer. 

The formulas of § 25 may be used to solve examples of the 
type given in § 16. 

Example 3. Given that sin 6 = -^\ and that tan d is negative, find the 
values of the other trigonometric functions. 

Since sin^ d -\- cos2 ^ = 1, it follows that cos 6 = ± ^|, but since tan is 
negative, lies in the second quadrant and cos d must be — i|. More- 
over, the relation tan = sin ^/cos gives tan =— y\. The reciprocals 
of these functions give sec ^ = — i|, esc = J/, ctn = — -y. 

EXERCISES 

1. Define secant of an angle ; cosecant ; cotangent. 

2. Are there any angles for which the secant is undefined ? If so, 
what are the angles ? Answer the same question for cosecant and co- 
tangent. 

3. Define versed sine ; coversed sine ; haversine. 

4. Complete the following formulas : 

sin2 4- cos2 = ? 1 + tan2 z= ? 1 + ctn2 0= ? tan ^ = ? 
Do these formulas hold for all angles ? 

5. In what quadrants is the secant positive ? negative ? the cosecant 
positive ? negative ? cotangent positive ? negative ? 

6. Is there an angle whose tangent is positive and whose cotangent is 
negative ? 

7. In what quadrant is an angle situated if we know that 

(a) its sine is positive and its cotangent is negative ? 
(6) its tangent is negative and its secant is positive ? 
(c) its cotangent is positive and its cosecant is negative ? 

8. Express sin2 ^ -f cos ^ so that it shall contain no trigonometric 
function except cos 0. 

9. Transform (1 -h ctn2 ^)csc so that it shall contain only sin 0. 

10. Which of the trigonometric functions are never less than one in 
absolute value ? 

11. Eor what angles is the following equation true : tan = ctn ? 

12. How many degrees are there in when ctn ^ = 1 ? ctn =— 1? 
sec ^ = \/2 ? CSC ^ = V2 ? 



Ill, § 27] SIMPLE TRIGONOMETRIC RELATIONS 37 

13. Determine from a figure the values of the secant, cosecant, and 
cotangent of 30°, 150^ 210°, 330°. 

14. Determine from a ligure the vahies of the secant, cosecant, and 
cotangent of 45^ 135°, 225°, 315°. 

15. Determine from a figure the values of the sine, cosine, tangent, 
secant, cosecant, and cotangent of 60°, 120°, 240°, 300°. 

16. Find 6 from the following equations. 

(tt) sin d = I. (0 tan ^ = — 1; 

(6) sin d=-l. ( j) ctn <9 = - 1. 

(c) cos d = |. (/c) tan ^ = 1. 

(d) cos ^ = - i. (0 ctni9 = 1. 

(e) sec = 2. (m) tan2 ^ = 3. 
(/) sec ^ = - 2. (n) sin ^ = 0. 
{g) CSC ^ = 2. (o) cos ^ = 0. 
Qi) CSC = -2. (p) tan ^ = 0. 

Prove the follovring identities and state for each the exceptional values 
of the variables, if any, for which one or both members are undefined : 



17. 


cos d tan e = 


: sin 0. 


18. 


sin ctn = 


cos^. 


19. 


1 + sin <9 _ 


cos (9 



cos 1 — sin 

20. sin2 __ cos2 = 2 sin2 0-1. 

21. (1 — sin2 0) csc2 = ctn2 0. 

22. tan ^ + ctn ^ = sec esc 0. 

23. [x sm0 -\-y cos 0^ + [xGos0—y sin 0Y = x2 + r/2. 

24. ^-^^ =cos^. 

tan -\- ctn 

25. 1 — ctn4 = 2 csc2 - csc^ 0. 

26. tan2 — sin2 = tan2 sin2 0. 

27. 2(1 H- sin 0) (1 -f cos (9) = (1 + sin (9 + cos 0)2. 

28. sin6 -h cos6 <9 = 1 - 3 sin2 cos2 ^. 

29. csc^ ^ csc^ :.2sec2g. 

CSC 0—1 CSC ^ + 1 

1 — tan _ Ctn 0^1 ^ 
1 -f- tan (9 ~ ctn -\- 1 

27. The Trigonometric Functions of 90° — 9. Figure 27 
represents angles 6 and 90° — 0, when is in each of the four 



38 



PLANE TRIGONOMETRY 



[III, § 27 



quadrants. Let OP be the terminal line of 6 and OP' the 
terminal line of 90° -0. Take OP' = OP and let {x,y) be 



T p' 



p y 



P' p y 



Fig. 27 




;-^ X 




the coordinates of P and (x', y') the coordinates of P\ 
in all four figures we have 

^' = y, y' = ^y r' = r. 

Hence 



Then 



sin (90° - 6>) = 2^^ = - = cos 6>, 



Also, 



cos (90° _ ^) = - = ^ = sin (9, 
r r 

tan (90° - ^) = ^ = ^ = ctn 0. 
x' y 



CSC (90°-^)= sec (9, 
sec (90° - ^) = CSC 6y 
ctn(90°-^)=tan(9. 

Definition. The sine and cosine, the tangent and cotangent, 
the secant and cosecant, are called co-functions of each other. 

The above results may be stated as follows : Any function 
of an angle is equal to the corresponding cofunction of the com- 
plementary angle,^ 

28. The Trigonometric Functions of 180° — 6. By draw- 
ing figures as in § 27, the following relations may be proved : 
sin (180° - ^) = sin 6, esc (180° -0)= esc 6, 

cos (180° - 6>) = - cos 6, sec (180° - 0) = - sec 6, 

tan (180° - 6>) = - tan 6, ctn (180° - 6>) = - ctn 6. 

The proof is left as an exercise. 

* Two angles are said to be complementary if their sum is 90°, regardless 
of the size of the ano^les. 



Ill, § 29] SIMPLE TRIGONOMETRIC RELATIONS 39 

29. The result of § 27 shows why it is possible to arrange 
the tables of the trigonometric functions with angles from 0° 
to 45° at the top of the pages and angles from 45° to 90° at 
the bottom of the pages. For example, since sin (90° — 0) = cos 0, 
the entry for cos 6 will serve equally well for sin (S0° — 0). 
As particular instances we may note sin 67° = cos 23°, tan 67° 
= ctn 23°, cos 67° = sin 23°. Verify these from the table. 

The result of § 28 enables us to find the values of the func- 
tions of an obtuse angle from tables that give the values only 
for acute angles. It will be noted that § 28 says that a7iy 
function of an obtuse angle is in absolute value equal to the same 
function of its supplementary angle but may differ from it in 
sign. 

Thus to find tan 137° we know that it is in absolute value 
the same as tan (180° - 137°) = tan 43° = 0.9325. But tan 137° 
is negative. Hence 

tan 137° = - 0.9325. 

Similarly, sin 137° = 0.6820. 

cos 137° = - 0.7314. 

EXERCISES 

Find the values of the following : 

tan 146°, sin 136^ cos 173^ tan 100^, cos 96^ sin 138% 
tan 98% sin 145% cos 168% cos 138°, tan 173% cos 157°. 



CHAPTER lY 



OBLIQUE TRIANGLES 

30. Law of Sines. Consider any triangle ABO with the 
altitude CD drawn from the vertex C (Fig. 28). 

On C 




In all cases we have sin ^4 = - , sin B = -> 

b a 

Therefore, dividing, we obtain 

sin A a 



- , or : 

b sm A 



(1) 



(2) 



sin B b ' sin A sin B 
If the perpendicular were dropped from B, the same argu- 
ment would give a/sin A = c/sin (7. Hence, we have 
d _ i> _ c 
sin A sin B sin C 

This law is known as the law of sines and may be stated as 
follows : Any two sides of a triangle are proportional to the 
sines of the angles opjjosite these sides. 

31. Law of Cosines. Consider any triangle ABC with the 
altitude CD drawn from the vertex C (Fig. 29). 
In Fig. 29 a 

AD = b cos A ; CD = b sin A ; DB = c — b cos A. 
In Fig. 29 b 

AD = — b cos A ; CD = b sin A ; DB = c — b cos A, 
In both figures 

a2 = DB'- + CD^. 

40 



IV, § 32] 
Therefore 



OBLIQUE TRIANGLES 

a- = c'2 - 2 he cos A + h"- cos2 A -|- ^2 siii2 ^ 
= o2 - 2 6c cos ^ + (cos2 ^ + sin2 A)b'^, 

C 



41 




whence 

a2 = 62 + c2 — 2 &c cos .4. 

The result holds also when J. is a right angle. Why ? 
Similarly it may be shown that 

62 = c2 + a^ - 2 ca cos B, 
c2 = a2 + 62-2a6cos C. 

Any one of these similar results is called the law of cosines. 
It may be stated as follows : 

Tlie square of any side of a triangle is equal to the sum of the 
squares of the other two sides diminished by twice the product of 
these two sides times the cosine of their included angle.^ 

32. Solution of Triangles. To solve a triangle is to find 
the parts not given, when certain parts are given. From 
geometry we know that a triangle is in general determined 
when three parts of the triangle, one of which is a side, 
are given.f Eight triangles have already been solved 
(§ 15), and we shall now make use of the laws of sines and 
cosines to solve oblique triangles. The methods employed 
will be illustrated by some examples. It will be found 
advantageous to construct the triangle to scale, for by so doing 
one can often detect errors which may have been made. 

* Of what three theorems in elementary geometry is this the equivalent ? 
t When two sides and an angle opposite one of them are given, the triangle 
is not always determined. Why ? 



42 PLANE TRIGONOMETRY 

33. Illustrative Examples. 



[IV, § 33 




Fui. :io 



Example 1. Solve the triangle ABC, given 
a = £76 A = 30° 20^ B = 60° 45', a = 276. 

Solution : 

C= 180°- (A + B)z= 180°- 91° 5' = 88° 55'; 



^ ^ g sin ^ ^ 276 sin 60° 45' ^ (276) (0.8725) ^ 47^3 g . 
sin^ sin 30° 20' 0.5050 



also 



^ ^ g sin C ^ 276 sin 88° o r>' ^ (276) (0.9998) ^ ^^^ ^ 
sin.l sin 30° 20' 0.5050 

•Check : It is left as an exercise to show that for these values we have 
c"-^ = a- -{■ b- — 2 ab cos C. 

Example 2. Solve the triangle ABC, given 
A = 30°, 5 = 10, a = 6. 

^6 Constructing the triangle ABC, we see that 

two triangles ABiC and AB^C answer the descrip- 




FiG. 31 



^ tion since 6 > a > altitude CB. 



Solution : Now 

sinB, ^ & ^ ^^ gij^ 5^ ^ 6^in^ ^q 833, 
sin A a a 

Bi = 56°. 5. 



whence 
But 

B2 = 180°- Bi = 180° - 56°.5 = 123°.5, 
and 
Ci = 180° - (^ + ^1) = 180° - 86°.5 = 93°. 5, 
C2 = 180° -{A-\- B2) = 180° - 153°. 5 = 26°.5. 
Now 

C2 _ sin C2 



a sin A sin A 0.500 



Also 



ci _ sin Ci _ osinCa _ (6)(0. 

— — 7 , or Ci — — ^ — — — 

a sm A sm J. 



0.500 



11. c 



Check : Ci^ = gS ^_ 52 _ 2 g6 cos Ci. 

143.5 = 36 + 100 +(2) (6) (10) (0.061) = 143.3. 

C22 = g2 + 62 - 2 ab cos C2. 
28.62 = 36 + 100 - (2) (6)(10) (0.895) = 28.60. 



IV, § 33] 



OBLIQUE TRIANGLES 



43 



Example 3. Solve the triangle ABC, given a = 10, h = 0, C = 40^. 
Solution : c'^ = a'^ -\- b^ — 2 ah cos C ^ 

Joy ' 



= 100 4- 36 -(120) (0.766)= 44.08. 
Therefore c = Q. 64. Now 



sm 



^ ^asin(;^ (10)(0.643) ^ ^^^g 



c 6.64 

ie. Jl = 104°. 5. Likewise, 





^.^^^^6sin_C^(6)(0.643)^ 

c 6.64 ' 



Check : A ^ B ^ C = 180°.0. 

Example 4. Solve the triangle ABC when 
a =1 7, 6=3, c = 5. 

From the law of cosines. 



cos.l = ^^ + ^-^-^^ = -l = -0.500, 

2 he 2 

2ac 14 

COS = ^^ + ^^^^ = 11 = 0.786. 
2 ah 14 



Therefore 



^ = 120°, 5 = 21^8, C = 38°.2. 
Check : A ■\- B -\- C = 180°.0 



EXERCISES 



1. Solve the triangle ABC, given 

(a) ^ = 30^ 5 = 70°, 

(5) A = 40°, B = 70°, 

(c) ^ = 4.5°.5, C = 68°.o, 

(d) l? = 60°.o, C = 44°20\ 

(e) a = 30, 6 = 54, C = 50° ; 

(/) 6 = 8, cf = 10, C = 60° ; 



a = 100 ; 
c = 110 ; 

5 = 40-; 
c = 20; 

(g) a = 10, 6 = 12, c = 14 ; 
(/i) a = 21, 6 = 24, c = 28. 



2. Determine the number of solutions of the triangle ABC when 



(a) A = 30°, 6 = 100, a = 70 

(6) ^=30°, 6 = 100, a = 100 

(c) ^ = 30°, 6 = 100, a = 50 

(d) ^ = 30°, 6 = 100, a = 40 



(e) A = 30'^, 6 = 100, a = 120 ; 
(/) A = 106°, 6 = 120, a = 16 ; 
(g) A= 90°, 6= 15, a= 14. 



44 



PLANE TRIGONOMETRY 



[IV, 33 



(c) A = 30°, a = 22, b= 34. 



One force 
rind the 



3. Solve the triangle ABC when 
(a) ^ = 37'-' 20', a = 20, 6 = 26; 
{b) ^ = 37^20', a = 40, 6 = 26; 

4. In order to find the distance from a point ^ to a point 5, a line 
AC and the angles CAB and ACB were measured and found to be 
300 yd., 60° 30^ 56° 10' respectively. Find the distance AB, 

5. In a parallelogram one side is 40 and one diagonal 90. The angle 
between the diagonals (opposite the side 40) is 25°. Find the length of 
the other diagonal and the other side. How many solutions ? 

6. Two observers 4 miles apart, facing each other, find that the angles 
of elevation of a balloon in the same vertical plane with themselves are 
60° and 40° respectively. Find the distance from the balloon to each 
observer and the height of the balloon. 

7. Two stakes A and B are on opposite sides of a stream ; a third 
stake C is set 100 feet from A^ and the angles J. C-B and CAB are observed 
to be 40° and 110°, respectively. How far is it from AtoB? 

8. The angle between the directions of two forces is 60°. 
is 10 pounds and the resultant of the two forces is 15 pounds 
other force.* 

9. Resolve a force of 90 pounds into two equal components whose 
directions make an angle of 60° with each other. 

10. An object B is wholly inaccessible and invisible from a certain 
point A, However, two points C and D on a line with A may be found 
such that from these points B is visible. If it is found that CD = 300 feet, 
^C = 120 feet, angle DCB = 70°, angle CDB = 50°, find the length AB, 

11. Given a, 6, A, in the triangle ABC. Show that the number of 
possible solutions are as follows : q 

^<90° 

( a<Cb sin A no solution, 
I 6 sin J. < a < 6 two solutions, 
\a^b ] 

[ a = bsinA \ 

^^90° 

\ ci^b no solution, 

I a > 5 one solution, 

12. The diagonals of a parallelogram are 14 and 16 and form an angle 
of 50°. Find the length of the sides. 

^ ^ * It is shown in physics that if the line segments AB 

and AC represent in magnitude and direction two 
forces acting at a point A, then the diagonal AD of the 
parallelogram ABCD represents both in magnitude and 
direction the resultant of the two given forces. 



one solution. 








IV, § 34] 



OBLIQUE TRIANGLES 



45 



13. Resolve a force of magnitude 150 into two components of 100 and 
80 and tind the angle between these components. 

14. It is sometimes desirable in surveying to extend a line such as AB 




in the adjoining figure. Show that this can be done by means of the 
broken line ABODE. What measurements are necessary ? 

15. Three circles of radii 2, 6, 5 are mutually tangent. Find the angles 
between their lines of centers. 

16. In order to find the distance between two objects A and B on op- 
posite sides of a house, a station C was chosen, and the distances CA 
= 500 ft., CB = 200 ft., together with the angle ACB = 65^30', were 
measured. Find the distance from A to B. 

17. The sides of a field are 10, 8, and 12 
rods respectively. Find the angle opposite the 
longer side. 

18. From a tower 80 feet high, two objects, 
A and B, in the plane of the base are found to 
have angles of depression of 13° and 10^ respec- 
tively ; the horizontal angle subtended by A and B at the foot C of the 
tower is 44°. Find the distance from A to B. 

34. Areas of Oblique Triangles. 

1. When two sides and the included angle are given. 
Denoting the area by S, we have from geometry 
C /S = I ch, 

but ^ = & sin ^ ; therefore 
(1) S = \ cb sin A. 

Likewise, 





Fig. 34 



S = ^ ah sin C and S ■■ 



i ac sin B. 



2. When a side and two adjacent angles are given. 

Suppose the side a and the adjacent angles B and C to be 

given. We have just seen that S = \ac sin B. But from the 

law of sines we have 

a sin (7 



c = 



sin A 



46 PLANE TRIGONOMETRY [IV, § 34 

Therefore 

^ _ g- « sin J5 » sin C 
2 sin^ 

But sin A = sin [180° - (j8 + (7)] = sin {B + G). Therefore 

o __ (X^ sin B sin C 
~2sin(^+(7)* 

3. Tr/ie7i ^/ie f^?'ee sides are given. 

We have seen that S = ^bc sin ^. Squaring both sides of 
this formula and transforming, we have 

^2 ^ ^ sin2 A = — (l- cos2^) 
4 4 

= |(l + cos^).|(l-cos^); 

whence, by the law of cosines, 

^c /^i , ^^ + c2 — a^\ hcf. 62 + c2 — a2' 



>S2 = ^ 1-f 






2V 2bG J 2\ 2 be 

2 6c + 62 ^ c2 - a2 2 6c - 62 -~ c2 + a2 



4 4 

6 + c + a 6 + c — a a— 6 + c a + 6- 



2 2 

which may be written in the form 

,^2 = s(s — a)(s — 6)(s — c), 
where 2s = a + 6 + c. Therefore, 



(2) S = ■Vs(s - a)(s - 6) (s - c). 

35. The Radius of the Inscribed Circle. If r is the radius 
of the inscribed circle, we have from elementary geometry, 
since s is half the perimeter of the triangle, S = 7^s', equating 
this value of S to that found in equation (2) of the last article 
and then solving for r, we get, 



-4 



(s — a)(s — b)(s — c) 
s 



IV, § 36] 



OBLIQUE TRIANGLES 



47 



EXERCISES 

Find the area of the triangle ABC, given ^ 

1. a = 25, 6 = 31.4, = 80^25'. 4. a = 10, 6 = 7, C=60°. 

2 6 = 24, c = 34.3, A = 60° 25'. 5. a = 10, 6 = 12, C = 60°. 

3. a = 37, 6 = 13, C = 40°. 6. a = 10, b = 12, C = 8^ 

7. Find the area of a parallelogram in terms of two adjacent sides 
and the inckided angle. 

8. The base of an isosceles triangle is 20 ft. and the area is 100/ VS 
sq. ft. Find the angles of the triangle. Ans. 30°, 30°, 120°. 

9. Find the radius of the inscribed circle of the triangle whose sides 
are 12, 10, 8. 

10= How many acres are there in a triangular field having one of its 
sides 50 rods in length and the two adjacent angles, respectively, 70° 
and 60° ? 

36. The Law of Tangents. For the work in the next 
chapter the formulas in this 
and the next article will be 
needed. 

Let CD be the bisector of 
the angle C of the A ABC. 
Through A draw a line II DC, 
meeting BC produced in E. 
Then CE = b. Why ? From 
A draw a line q _L DC meeting 

CB in F. At F draw a line r ± AF meeting AB in G. 
AE=p. 

Kow AACF is isosceles. Why? The angle ACE = ZA 

+ ZB and the bisector of Z.ACE is _L CD. Hence Z CAF 

==ZCFA = IZ{A + B). Moreover ZBAF=ZA-^Z{A 

+ B) = ^Z{A-B). 

B r^ 

Q 
A + B 

-I. 
r 




Let 



Now 



tan^i-±^ = i^ and tan — 

2 q 2 



tan 



2 



tan 



A-B 



48 


PLANE TRIGONOMETRY 


But 


p_BE_a + h ^^r.^ 
r BF a-b ^ ' 


Hence 


tan^ + ^ 

2 a + b 


tan^-^ «-^ 



[IV, § 36 



37. Angles of a Triangle in Terms of the Sides. Con- 
struct the inscribed circle of the triangle 
and denote its radius by r. If the perim- 
eter a + b + c = 2s, then (Fig. 36) 

AE = AF=s-a, 

A^s-c^F^ i. BD = BF=s-b, 

Fig. 3G Qjj ^ (J^ ^ ^ _ ^^ 

Then tanij. = -^^, tani5=-^^, tan|(7 = - ^ 




• a s — s — c 

where, from § 35, 



-V 



{s — a){s — b){s — c) ^ 



38. Solution of Triangles by Means of the Haversine. 

The haversine may be used advantageously in the solution of triangles, 
(1) when two sides and the included angle are given ; (2) when the 
three sides are given. The law of cosines gives 

2hav.d = l-cos^ = l _ ?>^ + c^ - ^2 

2 be 

2 6c 
or 4 6chav J. = a2 — (6 — c)2. 

1. If 5, c and A are given we may find a from the formula 

(1) fl2^(6~ c)2 -f 4 6chav^. 

Similar formulas give 62 or c^ in terms of a^c^B and a, 6, C respectively. 

2. If a, 6, c are given, we may find A from the formula 

(2) ^^^A^a'-ih-cY^is-l^is-c)^ 
^ ^ 4 be 6c 
Similar formulas will give B and C 



IV, § 38] OBLIQUE TRIANGLES 49 



Example 1. Given A = 04^ 


23' 


.4, 


6 = 55.12, r = .39.90. To find a. 


By formula (1) above : 






t 


b = 55.12 






be = 2199 


c = 39.90 






hav 94° 23'.4 = 0.0446 


(b-c)= 15.22 






6c hav ^= 1184 


(?j_c)2= 231.6 






4 6c hav .4 = 4736 


4 be hav A = 4736 









a2 = 4968 
a = 70.49 
Example 2. Given a = 4.51, b = 6.13, c = 8.16. Find A, B, C. 



a2 




hav ^=i^^ = 0.0811 A= 33^05 
200.1 



4 6c = 200.1 

^^- 3^-58 hav 5 =2^^ = 0.1648 B= 47^54 
(c-q)2^ 13.32 147.21 

52_(c-a)2= 24.26 

ac= 36.80 

4ac = 147.21 

c2= 66.59 have = -^5:51 =0.5785 C^ 99^02^ 
(6-«)2 = _2^ 110.58 ^^^^^, ^^^r^ 

c2^^b-ay= 63.97 
ab= 27.646 
4a6 = 110.58 

EXERCISES 

Solve the following triangles : 

1. a = 62.1, 6 = 32.7, c = 47.2. 

2. ^ = 37° 20', 6 = 2.4, c = 4.7. 

3. B = 12r.32', a = 27.9, c = .35.8. 

4. a = 3.2, 6 = 5.7, c = 6.5. 

5. C = 72°21'.4, a = 314.1, 6 = 427.3. 

6. a = 346.1, 6 = 425.8, c = 562.3. 



CHAPTER V 



LOGARITHMS 

39. The Invention of Logarithms. The extensive numeri- 
cal computations required in business, in science, and in engi- 
neering were greatly simplified by the invention of logarithms 
by John Napier, Baron of Merchiston (1550-1617). By means 
of logarithms w^ are able to replace multiplication and division 
by addition and subtraction, processes which we all realize are 
more expeditious than the first two. 

If we consider the successive integral powers of 2 



(1) 



Exponent x 


1 


2 


3 


4 


5 


6 


7 


Result 2^ . . 


2 


4 


8 


16 


32 


64 


128 




Exponent x . 


8 


9 


10 


11 


12 


etc. 


A. P. 


Result 2^ . . 


256 


512 


1024 


2048 


4096 


etc. 


G. P. 



we see that the results form a geometric progression (G. P.) 
and the exponents an arithmetic progression (A. P.). We 
know from elementary algebra that 



and 



^n _ ^m 



■ = X"" 



Hence if we wish to multiply two numbers in our G. P. e.g, 
4 X 8, we merely have to add the corresponding exponents 2 
and 3 and under the sum 5 find the desired product 32. Sim- 
ilarly, if we wish to divide e.g. 4096 by 128, we merely have to 
subtract the exponent corresponding to 128, from that cor- 

50 



\-. § 39] 



LOGARITHMS 



51 



responding to 4096 and under their difference 5 we find ^he 
desired quotient 32. 

To make the above plan at all useful it is evident that our 
table must be expanded so as to contain more numbers. First 
we can expand our table so that it will contain numbers less 
than 2, by subtracting 1 successively from the numbers in the 
A. P. and by dividing successively by 2 the numbers in the 
G.P. 



(2) 



In the second place we may find new numbers by inserting 
arithmetic means and geometric means. Thus, if we take the 
following portion of the preceding table 



-5 


-4 


-3 


-2 


-1 





1 


2 


3 


4 


5 


6 


7 


0.03125 


0.0625 


0.125 


0.25 


0.5 


1 


2 


4 


8 


16 


32 


64 


128 



-2 


- 1 





1 


2 


3 


4 


i 


i 


1 


2 


4 


8 


16 



and insert between every two successive numbers of the upper 
line their arithmetic, and between every two successive num- 
bers of the lower line their geometric mean, we obtain the 
table 



-2 


-f 


-1 


-i 





i 


1 


1 


2 


. f 


3 


i 


4 


i 


iV2 


i 


-J-V2 


1 


V2 


2 


2V2 


4 


4V2 


8 


8V2 


16 



(3) 



If the radicals are expressed approximately as decimals, this 
table takes the form 



-2.0 


-1.5 


-1.0 


-0.5 



1.00 


0.5 


1.0 


1.5 


2 


2.5 


3 


3.5 


4 

16 


0.25 


0.35 


0.50 


0.72 


1.41 


2.00 


2.83 


4.00 


5.66 


8.00 


11.31 



52 PLANE TRIGONOMETRY [V, § 39 

By continuing this process we can make any number appear 
in the G. P. to as high a degree of approximation as we desire. 
To prepare an extensive table, which gives values at small inter- 
vals, is quite laborious. However, it has been done, and we 
have printed tables so complete that actual multiplication of 
any two numbers can' be replaced by addition of two other 
numbers. We shall soon learn how to use such tables. 

40. Definition of the Logarithm. The logarithm of a 

number ^ to a base 6 (6 > 0, =5!^ 1) is the exponent x of 
the power to which the base h must be raised to produce the 
number J^, 
That is, if 

&- = AT, 
then 

These two equations are of the highest importance in all work 
concerning logarithms. One should keep in mind the fact 
that if either of them is given, the other may always be 
inferred. 

The numbers forming the A. P. in tables 1, 2, and 3 of § 39 
are the logarithms of the corresponding numbers in the G. P., 
the base being 2. From table 3 we have 2^ = 4\/2 which says 
log2 4V2 = |. 

EXERCISES 

1. When 3 is the base what are the logarithms of 9, 27, 3, 1, 81, 1, 

1 

2. Why cannot 1 be used as the base of a system of logarithms ? 

3. When 10 is the base what are the logarithms of 1, 10, 100, 1000 ? 

4. Find the values of x which will satisfy each of the following 
equalities : 

(a) logs 27 = X. (d) logo a = X. {g) logs x = 6. 

(6) log^ 3 = 1. (e) loga 1 = X. {K) logas x = J. 

(c) log^ 5 = i. (/) loga -h = ^' (0 logo.ooi x = 2. 



V, § 41] LOGARITHMS 53 

t 

5. Find the value of each of the following expressions : 

(a) log 2 16. (c) loge^l^ (e) log25 125. 

(6) log343 49. (d) log2Vi6. (/) log2^V. 

41. The Three Fundamental Laws of Logarithms. From 
the laws of exponents we derive, the following fundamental 
laws. 

I. The logarithm of a product equals the sum of the logarithms 
of its factors. Symbolically, 

logft MN = log5 M + logft A^. 

Proof. Let log^ M = x, then b"" = M. Let log^ ^= !/) then 
6'^ = X. Hence we have 3fJSf = &^+^, or 

log, MX =x + y, i.e. log, JAY = log, M + log, JV. 

II. TJie logarithm of a quotient equals the logarithm of the 
dividend minus the logarithm of the divisor. Symbolically, 

log6^= logft M — logfe N. 

N 

Proof. Let log, M = x, then 6^ = M. Let log, X^ y, then 
6'^ = X. Hence we have M/X = 6^~^, or 

M M 

^ogt,^— = x-y, i.e. log,^ = log, Jf - log, Y". 

III. The logarithm of the pth power of a number equals p 
times the logarithm of the number. Symbolically 

logft MP = p logft M. 

Proof. Let log, M = x, then b"" = M. Kaising both sides 
to the ^th power, we have b^"" = M^. Therefore 

log, M^ =2)x = p log, .v. 

Prom law III it follows that the logarithm of the real jjositive 
nth root of a number is one nth of the logarithm of the member. 



54 PLANE TRIGONOMETRY [V, § 41 

EXERCISES 

1. Given logio 2 = 0.3010, logio 3 = 0.4771, logio 7 = 0.8451, find the 
value of each of the following expressions : 

(a) logio 6. . (/) logio 5. 

[Hint: logio 2x3= logio 2 + logio 3.] [Hint: logio 5 = logio V--] 

(b) logio 21.0. (9) logio 15a 

(c) logio 20.0. W logio Vl4. 

(d) logio 0.03. (i) logio 49^ 

(e) logio |. 0") logio V24. 75. 

2. Given the same three logarithms as in Ex. 1, find the value of each 
of the following expressions : 

^^\ 1^^ 4 X 5 X 7 ^7.N i^rr ^ X 3 X 20 . . , „ 2058 

(d) logio (2)25. (e) logio (3)3(5)6, (/) logio(23)(i). 

42. Logarithms to the Base 10. Logarithms to the base 10 
are known as common or Briggian logarithms. Proceeding as 
in § 39 we can show that lO^-^^^^^ = 2, i.e. logio ^ = 0.3010. Let 
us multiply both members of the equation lO^-^^^^ = 2 by 10, 10^, 
10^, etc. and notice the effect on the logarithm. 
100.3010 ^ 2 logio 2 = 0.3010 

10 3«io = 20 logio 20 = 1.3010 

1Q2.3010 ^ 200 logio 200 = 2.3010. 

It should be clear from this example that the decimal part of 
the logarithm (called the mantissa) of a number greater than 1 
depends only on the succession of figures composing the num- 
ber and not on the position of the decimal point, while the in- 
tegral part (called the characteristic) depends simply on the 
position of the decimal point. Hence it is only necessary to 
tabulate the mantissas, for the characteristics can be found by 
inspection as the following considerations show. 

Since 
100 = 1, 10^ = 10, 102 = 100, 10^ = 1000, 10^ = 10,000, etc. 
we have logio 1 = 0, log^o 10 = 1, log^o 100 = 2, 

logio 1000 = 3, logio 10,000 = 4, etc. 



V, § 42] LOGARITHMS 55 

It follows that a number with one digit (9^= 0) at the left of the 
decimal poiut has for its logarithm a number equal to + a 
decimal ; a number with tv:o digits at the left of its decimal 
point has for its logarithm a number equal to 1 + a decimal ; a 
number with three digits at the left of the decimal point has 
for its logarithm a number equal to 2 + a decimal, etc. We 
conclude, therefore, that the characteristic of the common loga- 
rithm of a number greater than 1 is one less than the number of 
digits at the left of the decimal point. 

Thus, logio 456.07 = 2.65903. 

The case of a logarithm of a number less than 1 requires 
special consideration. Taking the numerical example first con- 
sidered above, if logio 2 = 0.30103, we have logio 0.2 = 0.30103-1. 
Why? This is a negative number, as it should be (since the 
logarithms of numbers less than 1 are all negative, if the 
base is greater than 1). But, if we were to carry out this 
subtraction and write logio 0-- = — 0.69897 (which would be 
correct), it would change the mantissa, which is inconvenient. 
Hence it is customary to write such a logarithm in the form 
9.30103 - 10. 

If there are n ciphers immediately following the decimal 
point in a number less than 1, the characteristic is — n— 1. 
For convenience, if n < 10, we icrite this as (9 — n) — 10. TJiis 
characteristic is ivntten in two parts. The first part 9 — n is 
icritten at the left of the mantissa and the — 10 at the right. 

In the sequel, unless the contrary is specifically stated, we 
shall assume that all logarithms are to the base 10. We may 
accordingly omit writing the base in the symbol log when there 
is no danger of confusion. Thus, the equation log 2 = 0.30103 
means logio 2 = 0.30103. 

To make practical use of logarithms in computation it is 
necessary to have a conveniently arranged table from which 
we can find (a) the logarithm of a given number and (b) the 
number corresponding to a given logarithm. The general 



56 PLANE TRIGONOMETRY [V, § 42 

principles governing the use of tables will be explained by the 
following examples [Tables, pp. 110, 111]. 

Example 1. Find log 42.7. 

The characteristic is 1. In the column headed N (p. 110) we find 42 
and if we follow this row across to the column headed 7, we read 6304, 
which is the desired mantissa. Hence log 42.7 = 1.6304. 

Example 2. Eind log 0.03273. 

The characteristic is 8 — 10. The mantissa cannot be found in our 
table, but we can obtain it by a process called interpolation. We shall 
assume that to a small change in the number there corresponds a propor- 
tional change in the mantissa. Schematically we have 

Number Mantissa 

^ r3270 -> 5145- 

^13273 — > ? 14 = difference 

3280 -> 5159 



difference = 10 



Our desired mantissa is 5145 + j3_ . 14 = 5149. Hence log 0.03273 
= 8.5149 - 10. 

Example 3. Find x when logic = 0.8485. 

We cannot find this mantissa in our table, but we can find 8482 and 
8488 which correspond to 7050 and 7060 respectively. Reversing the 
process of example 2, we have schematically 







Number 


Mantissa 






"7050 ^r- 


84821 1 
8485J 




Difference = 


10 


? •<r- 


6 = difference 






7060 <- 


8488 





Hence the significant figures in our required number are 7050 + | . 10 
= 7055. Since the characteristic is the required number is 7.055. 

EXERCISES 

1. Eind the logarithms of the following numbers from the table on 
pp. 110, 111 : 482, 26.4, 6.857, 9001, 0.5932, 0.08628, 0.00038. 

2. Eind the numbers corresponding to the following logarithms : 
2.7935, 0.3502, 7.9599 - 10, 9.5300 - 10, 3.6598, 1.0958. 

43. Use of Logarithms in Computation. The way in 

which logarithms may be used in computation will be suffi- 
ciently explained in the following examples. A few devices 
often necessary or at least desirable will be introduced. The 



V, § 43] LOGARITHMS 57 

latter are usually self-explanatory. Eeference is made t6 
them here, in order that one may be sure to note them when 
they arise. The use of logarithms in computation depends, of 
eou3-se, on the fundamental properties derived in § 41. 

Example 1. Find the value of 73.26 x 8.914 x 0.9214. 
We find the logarithms of the factors, add them, and then find the 
number corresponding to this logarithm. The work may be arranged as 
follows : 

Numbers Logarithms 

73.26 (-^) 1.8649 

8.914 (-^) 0.9501 

0.9214 (->) 9.9645 - 10 

12.7795 - 10 
Product = 601.9 ^ns. (<-) 2.7795 

Example 2. Eind the value of 732.6 -f- 89.14. 

Numbers Logarithms 

732.6 (->) 2.8649 

89.14 (^) 1.9501 

Quotient = 8.219 Ans, (<-) 0.9148 

Example 3. Eind the value of 89.14 -4- 732.6. 

Numbers Logarithms 

89.14 (->) 11.9501-10 

732.6 (->) 2.8649 

Quotient = 0.1217 ^ns. (<~) 9.0852 - 10 

763.2 X 21.63 



Example 4. Eind the value of 



.i 



Whenever an example involves several different operations on the 
logarithms as in this case, it is desirable to make out a blank form. When 
a blank form is used, all logarithms should be looked up first and entered 
in their proper places. After this has been done, the necessary opera- 
tions (addition, subtraction, etc.) are performed. Such a procedure 
saves time and minimizes the chance of error. 

EORM 

Numbers Logarithms 

763.3 (-^) 

21.63 (-^) ( + ) 

product 

986.7 (-» (-) 

. . . . Ans. (<—) 



58 



PLANE TRIGONOMETRY 



[V, §43 



Form Filled In 
Numbers 

763.2 (->) 

21.63 (->) 

product 

986.7 (->) 

16.73 Ans. (<-) 



Example 5. Find (1.357)^ 
Numbers 
1.357 



(-» 



(1.357)5 = 4.602 Ans. (<-) 

Example 6. Find the cube root of 30.11. 
Numbers 

30.11 (-^) 

\/30.11 =3.111 Ans. (-e-) 



Logarithms 
2.8826 
1.3351 
4.2177 
2.9942 
1.2235 



Logarithms 
0.1326 
0.6630 



Logarithms 
1.4787 
0.4929 



Example 7. Find the cube root of 0.08244. 

Numbers Logarithms 

0.08244 (-^) 28.9161 - 30 

v/0.08244 = 0.4352 Ans. (^ 9.6387-10 



EXERCISES 

Compute the value of each of the following expressions using the table 
on pp. 110, 111. 



1. 


34.96 X 4.65. 


2. 


518.7 X 9.02 X .0472. 


3 


0.5683 




0.3216 


4. 


5.007 X 2.483 
6.524 X 1.110 


6. 


(34.16 X .238)2. 


6. 


8,572 X 1.973 x (.8723)2. 


7 


'/ 648.8 
"V (21.4)2 




8. 


J1379. 
>'2791 



9. 



10. 



4 



'2. 8076 X 3.184 ^ 
(2.012)3 



2941 X 17.32 
2173 X 18.75* 



11. 



VO. 00732 ^ 
\/735 

12. (20.027)i 

13. 2100. 



14- ^1502^ 100^. 
15. ro n9.7.«?.^^i 
16. 



(0.02735) ^ 

V3275 



(2.01)^ 



V, § 44] LOGARITHMS 59 

M 1 ' 

44. Cologarithms. Since — and M are equivalent, 

we may in a logarithmic computation, add the logarithm of 
— instead of subtracting log N, The logarithm of — is 

called the cologarithm of N, Therefore 

colog N = log 1/^ = log 1 — log N = — log N, 

since log 1 is zero. 

We write cologarithms, like logarithms, with positive man- 
tissas. Therefore the cologarithm is most easily found by sub-^ 
tracting the logarithm from zero, written in the form 10.0000 
-^10. 

Example. Find the colog 27.3. 

10.0000 - 10 
lo g 27. 3 rr 1.4362 
colog 27. 3= 8.6638-10 

The cologarithm can be written down immediately by subtracting the 
last significant figure of the logarithm from 10 and each of the others 
from 9. If the logarithm is positive the cologarithm is negative and 
hence — 10 is affixed. 

There is no gain in using cologarithms when we have a quotient of two 
numbers. There is an advantage when either the numerator or denomi- 
nator contains two or more factors, for we can save an operation of addi 
tion or subtraction. Let us solve Ex. 4, § 43, using cologarithms. 



1. Find the value of - 


986.7 




Numbers 




Log 




763.2 


-^ 


2.8826 




21.63 


-> 


1.3351 




986.7 


-> 


(colog) 7.0058 - 


-10 


16.73 


<- 


1.2235 





EXERCISES 

Compute the value of each of the following expressions, using cologa- 
rithms. 



/ 2.80 X 37.6 / 

>'4.96 X 23.3* ' V: 



J ^/^.uv ^ t^.-v. « ^, 97.63x876.5 



4.96 X 23.3 >2876 x 3.4 x 2.987 



60 PLANE TRIGONOMETRY [V, §44 

3 5 5 V3275 

• 7x8x9x27.6' ' (2.01)*(1.76)io* 

3^^ . ^ 1293 X 12 7 X 5 



4. 



^'''^' ' (l + ^V3)(760 + 8) 



MISCELLANEOUS EXERCISES 

1. What objections are there to the use of a negative number as the 
base of a system of logarithms ? 

2. Show that a^^Sa' = x. 

3. Write each of the following expressions as a single term : 

(a) log X + logy — log z. (6) 3 log x — 2 log 2/ + 3 log z. 

(c) 3 log a — log (x + ?/) — I log (ex + c^) + log Vw + x. 

4. Solve for x the following equations : 

(a) 2 log2 X + log2 4 = 1. (c) 2 logio x — S logio 2 = 4. 

(6) logs X - 3 logs 2 = 4. (d) 3 log2 x + 2 log2 3 = 1, 

5. How many digits are there in 2^5 ? 31^^ ? 3^2 x 28 ? 

6. Which is the greater, (|J)ioo or 100 ? 

7. Find the value of each of the following expressions : 

(a) log6 35. (6) logs 34. (c) logy 245. (d) logi3 26. 

8. Prove that log^ a • log^ 6 = 1. 

9. Prove that 

loga ^ + ^ -^^ ~ ^ = 2 log, [X 4- Vi^"=l]. 

X - \/x2 - 1 

10. The velocity v in feet per second of a body that has fallen s feet 
is given by the formula v = V64.3 s. 

What is the velocity acquired by the body if it falls 45 ft. 7 in. ? 

11. Solve for x and ?/ the equations : 2^ = 16^^, x + 4 ?/ = 4. 



CHAPTER VI 

LOGARITHMIC COMPUTATION 

45. Logarithmic Computation. In the last chapter a few 
examples of the use of logarithms in computation were given 
in connection with a four-place table. Such a table suffices 
for data and results accurate to four significant figures. When 
greater accuracy is desired we use a five-, six-, or seven-place 
table. 

No subject is better adapted to illustrate the use of logarith- 
mic computation than the solution of triangles, which we shall 
consider in some detail. Five-place tables and logarithmic 
solutions ordinarily are used at the same time, since both tend 
toward greater speed and accuracy. 

46. Five-place Tables of Logarithms and Trigonometric 
Functions. The use of a five-place table of logarithms differs 
from that of a four-place table in the general use of so-called 
" interpolation tables '' or " tables of proportional parts,'' to 
facilitate interpolation. Since the use of such tables of pro- 
portional parts is fully explained in every good set of tables, 
it is unnecessary to give such an explanation here. It will be 
assumed that the student has made himself familiar with their 
use."^ 

In the logarithmic solution of a triangle we nearly always 
need to find the logarithms of certain trigonometric functions. 
For example, if the angles A and B and the side a are given, 
we find the side b from the law of sines given in § 30, 

, a sin 5 

^ = — — T' 

suiA 

* For this chapter, such a five-place table should be purchased. See, for 
example. The Macmillan Tables, which contain all the tables mentioned 
here with an explanation of their use. 

61 



62 PLANE TRIGONOMETRY [VI, § 46 

To use logarithms we should then have to find log a, log (sin B) 
and log (sin A), With only a table of natural functions and a 
table of logarithms at our disposal, we should have to find first 
sin Ay and then log sin A. For example, ii A = 36° 20', we 
would find sin 36° 20' = 0.59248, and from this would find log 
sin 36° 20' = log 0.59248 = 9.77268 - 10. This double use of 
tables has been made unnecessary by the direct tabulation of the 
logarithms of the trigonometric functions in terms of the angles. 
Such tables are called tables of logarithmic sines, logarithmic 
cosines, etc. Their use is explained in any good set of tables. 
The following exercises are for the purpose of familiarizing 
the student with the use of such tables. 

EXERCISES 

1. Find the following logarithms : * 

(a) log cos 27° 40'.5. (d) log ctn 86° 53'.6. 

(h) log tan 85° 20'. 2. (e) log cos 87° 6'.2. 

(c) logsin45°40'.7. (/) log cos 36° 53'. 3. 

2. Find A^ when 

(a) log sin A = 9.81632 - 10. (d) log sin A = 9.78332 - 10. 

(6) log cos^ = 9.97970 - 10. (e) log ctni^ = 0.70352. 

(c) log tan A = 0.45704. (/) log tan J ^ = 9.94365 - 10. 

3. Find^,iftan^=^IM22L§M10. 

87325 

4. Given a triangle ABC, in which ZA = 32°, ZB = 27°, a = 5.2, find 
6 by use of logarithms. 

47. The Logarithmic Solution of Triangles. The effective 
use of logarithms in numerical computation depends largely on 
a proper arrangement of the work. In order to secure this, 
the arrangement should be carefully planned beforehand by 
constructing a blank form, which is afterwards filled in. More- 
over, a practical computation is not complete until its accuracy 
has been checked. The blank form should provide also for a 
good check. Most computers find it advantageous to arrange 

* Five-place logarithms are properly used when angles are measured to the 
nearest tenth of a minute. For accuracy to the nearest second, six places 
should be used. 



VI, § 48] LOGARITHMIC COMPUTATION 63, 

the work in two columns, the one at the left containing the 
given numbers and the computed results, the one on the right 
containing the logarithms of the numbers each in the same 
horizontal line with its number. The work should be so 
arranged that every number or logarithm that appears is 
properly labeled ; for it often happens that the same number 
or logarithm is used several times in the same computation and 
it should be possible to locate it at a glance when it is wanted. 
The solution of triangles may be conveniently classified 
under four cases : 

Case I. Given two angles and one side. 

Case II. Given two sides and the angle opposite one of the 
sides. 

Case III. Given tivo sides and the included angle. 

Case IV. Given the three sides. 

In each case it is desirable (1) to draw a figure representing 
the triangle to be solved with sufficient accuracy to serve as a 
rough check on the results ; (2) to write out all the formulas 
needed for the solution and the check ; (3) to prepare a blank 
form for the logarithmic solution on the basis of these 
formulas ; (4) to fill in the blank form and thus to complete 
the solution. 

We give a sample of a blank form under Case I ; the student 
should prepare his own forms for the other cases. 

48. Case I. Given Two Angles and One Side. 

Example. Given : a = 430.17, ^ = 47° 13'.2, 5=52° 29'.5. (Fig. 37.) 
To find: C, b, c. 

Formulas : q 

C=180°-(^ + i^), 

6=—^ sin 5, 
sin^ 

- - ^ sin C. 



sin^ 



Check (§ 36) : £^ = t^nlJC-B) 

^^ ^ c + 6 t.z,n\{C+B) ^ Fig. 37 




64 



PLANE TRIGONOMETRY 



[VI, § 48 



The following is a convenient blank form for the logarithmic solu- 
tion. The sign ( + ) indicates that the numbers should be added ; the 
sign (— ) indicates that the number should be subtracted from, the one 
just above it. 



A = 
Ai- B = 

C=. . 

a = . . 
sin A = sin 

a/sin A 
sin B = sin 
6= . . 



Numbers 



179^ 60'.0 



Logarithms 



a/sin A 
sin C = sin 



c-\- b = 



C- B= . , 
C+ B= . . 

tan^(C— l?)i= tan 
tan^(C+ 5)= tan 





(- 



■^) (+) 
^) 

-^) (+) 

Check 

-» (-) 



(1) 



(Logs (1) and (2) 

►) should be equal 

>■) ( — ) for check.) 

(2) 
FilUng in this blank form, we obtain the solution as follows. 
Numbers Logarithms 

A= 47°13'.2 
B= 52°29\6 
A-{- B= 99°42'.8 
179°60'.0 



C= 80° 17'. 2 



a = 430.17 


(-^) 


2.63364 




sinJl = sin 47° 13'. 2 


(-» 


(-) 9.86567- 


10 


a/sin A 




2.76797 




sinB = sin52°29'.6 


(-» 


( + ) 9.89943- 


-10 


b = 464.94 Ans. 


C^) 


2.66740 





Check* 



VI, § 49] LOGARITHMIC COMPUTATION 65 , 

a/sin A 2.76797 

sin C = sin BOM 7 '.2 (->) (+) 9.99373 -10 
c = 677.70 Ans. {<-) 2.76170 

Check 
c-b= 112.76 (->) 2.05215 

c + 6 = 1042.64 (->) (-) 3.01813 

9.03402-10] 
C-B= 27M7'.6 
C+ ^ = 132° 46'. 8 
tan ^(C- B)= tan 13° 53'.8 (->) 9.39342 - 10 

tan i ( C + ^) = tan 66° 23'. 4 (->) (-) 0.35942 

9.03400-10. 

EXERCISES 

Solve and check the following triangles ABC : 

1. a = 372.5, A = 25° 30', B = 47° 50'. 

2. c =327.85, ^ = 110°52'.9, 5 = 40°31'.7. Ans. C = 28°35'.4, 

a = 640.11, 6 = 445.20. 

3. a = 53.276, 4 = 108° 50'.0, C = 57°13'.2. 

4. 6=22.766, 5 = 141° 59'.1, C = 25° 12'.4. 

5. 6 = 1000.0, J5 = 30°30'.5, C = 50°50'.8. 

6. a = 257.7, ^ = 47° 25', 5 = 32° 26'. 

49. Case II. Given Two Sides and an Angle Opposite 
One of Them. 

If A, a, b are given, B may be determined from the relation 

(1) sm ^ = 



a 



If log sin ^ = 0, the triangle is a right triangle. Why ? 

If log sin J5 > 0, the triangle is impossible. Why ? 

If log sin ^ < 0, there are two possible values, Bi, B^ of B, 
which are supplementary. 

Hence there may be two solutions of the triangle. (See 
Example.) 

No confusion need arise from the various possibilities if the 
corresponding figure is constructed and kept in mind. 

It is desirable to go through the computation for log sin B 

* A small discrepancy in the last figure need not cause concern. Why ? 

F 



66 



PLANE TRIGONOMETRY 



[VI, § 49 



before making out the rest of the blank form, unless the data 

obviously show what the conditions of the problem actually 

are. 

Example 1. Given : A = 46° 22^2, a = 1.4063, b = 2.1048. (Fig. 38.) 
To find: B, C, c. 

Formula: smB = ^^^. 




Fig. 38 



Numbers 
6 = 2.1048 (- 

sin ^ = sin 46^ 22'. 2 (- 
bsinA 

a = 1.4063 (- 

sin B (- 



Logarithms 
0.32321 

/) ( + ) 9.85962 - 10 

0.18283 
(-) 0.14808 
-) 0.03475 



Hence the triangle is impossible. Why ? 

Example 2. Given : a = 73.221, b = 101.53, A = 40° 22'.3. (Fig. 39.) 
To find: B, C, c. 

Formula: sin^= ^^"^^ . 



Logarithms 
2.00660 

*^) ( + ) 9.81140 - 10 



Numbers 
b = 101.53 (- 

sin ^ = sin 40° 22'. 3 (- 
6sinJL 11.81800-10 

(T = 73.221 (->) (-) 1.86464 

sin B 9.95336 - 10 



The triangle is therefore possible and 
has two solutions (as the figure shows) . 
We then proceed with the solution as 
follows : 

We find one value Bi of B from 
the value of log sin B. The other 
value B2 of B is then given by B2 = 
180° - J5i. 




VI, § 49] LOGARITHMIC COMPUTATION 67 



Other formulas : 

C= 180^ -(.-l-f B). 

a sin C 
c = 

sin A 

Chbck: 'L^^'^IiiiC^^_. 

c -\- b tan^(C+ B) 

Numbers Logarithms 

sin B 9.95336 - 10 

Bi= 63°55'.2 
179° 60\0 
B2= 116° -i'.S 

A-\- Bi = 104°17'.5 

179° 60^0 

Ci= 75° 42'. 5 

a (~>) 1.86464 

sin^ (-^) (-) 9.81140 -10 

a/ sin A 2.05324 

sin Ci = sin75" 42.5 (->) ( + ) 9.98634 - 10 
Ci = 109.54 (<-) 2.03958 

Ci-6= 8.01 (->) 0.90363 

CiH- 6 = 211.07 (-^) (-) 2.32443 

8.57920-10 1 



Ci- Bi= 11° 47'. 3 
Ci + jBi = 139° 37'.7 



i 
! 

tanirCi - Bi)= tan5° 53'. 6 (->) 9.01377 - 10 [ ^^^^^* 

tan ^( Ci + Bi) = tan 69° 48'. 8 (->) 0.43455 } 

8.57922 - 10 J 

One solution of the triangle gives, therefore. B = 63° 55'. 2, C = 75° 42'. 5, 
c = 109.54. 

To obtain the second solution, we begin with Bo = 116° 4'. 8. We find 
C2 from C2 = 180° -(J. + ^2); i-e. C2 = 23° 32'. 9. The rest of the com- 
putation is similar to that above and is left as an exercise. 



EXERCISES 

1. Show that, given A^ a, 6, if A is obtuse, or if A is acute and a > &, 
there cannot be more than one solution. 

Solve the following triangles and check the solutions : 
2.0 = 32.479, 6 = 40.176, ^ = 37° 25M. 



68 



PLANE TRIGONOMETRY 



tVI, § 49 



3. 6 = 4168.2, c = 3179.8, B = 61°2VA. 

4. a = 2.4621, 6 = 4.1347, 5 = 101° 37^3. 

5. a = 421.6, c = 532.7, A = 49° 21^8. 

6. a = 461.5, c = 121.2, C=22°31'.6. 

7. Find the areas of the triangles in Exs. 2-5. 

50. Case HI. Given Two Sides and the Included Angle. 

Example. Given: a=214.17, 6=356.21, 

B C = 62°21'.4. (Fig. 40.) 

To find : A, B, c. 
A/ \ Formulas : 




Numbers 
b-a = 142.04 
6 -f a = 570.38 
(6 - a)/(6 + a) 

tan i(5 -i-A)= tan 58° 49'. 3 

tani(5- J.)=tan22°22'.2 

,'.A= 36°27M 

B= 81°11'.5 

a = 214.17 

sinJ. = sin36°27M 

a/sin A 

sin C = sin62°21'.4 
c = 319.32 Ans. 
Check by finding log (6/sin B). 



tan 1 


(i?-^) = ^^tani(^ + ^); 
6 + ci 




B-{-A = 180° - C = 117° 38^6 




a sin C _ 




sin^ 




Logarithms 


(-» 


2.15241 


(-» 


(-) 2.75616 




9.39625 - 10 


(^) 


( + ) 0.21817 


«~) 


9.61442 - 10 


J.ns. 




^?is. 




(->) 


2.33076 


(-» 


(-) 9.77389-10 




2.55687 


(-» 


( + ) 9.94736-10 


(^) 


2.50423 



EXERCISES 

Solve aud check each of the following triangles : 

1. a = 74.801, 6 = 37.502, C = 63°35'.5. 

2. a =423.84, 6 = 350.11, C = 43° 14'.7. 

3. 6 = 275, c = 315, A = 30° 30/. 

4. a =150.17, c = 251.09, jB = 40°40'.2. 

5. a = 0.25089, 6 = 0.30007, C = 42° 30' 20^ 

6. Find the areas of the triangles in Exs. 1-5. 



VI, § 51] LOGARITHIMIC COMPUTATION 



69 



51. Case IV. Given the Three 
Sides. 

Example. Given: a = 261.62, 
b =: 322.42, 
c = 291.48. 
To find: A, B, C. 
Formulas : 

s= i(a+ 6 + 0- 



-4 



(s — g) (s — 6) (s — c) 




tan ^ A : 



tan J 5 : 



s — a 
Check : ^ + 5 + C = 180° 

Numbers 

a = 261.62 

6 = 322.42 

c ^ 291.48 

2s = 875.52 

s = 437.76 

s - a = 176.14 

s-b = 115.34 

s - c = 146.28 



r 



tan ^ C = - 



Logarithms 



8 = 437.76 (Check). *(- 
r2 



2.24586 

2.06198 

( + ) 2.16518 

6.47302 
.^) (-) 2.64124 

3.83178 



r 
s — a 
tan J J. : 

r 
s-b 



: tan25°4M 



tan 35° 32'. 4 



i<-) 



«-) 



r = 
s — c = 
tan i C = tan 29° 23 .4+ (<-) 
A= 50° 8'.2 Ans. 
B= 71° 4^8 ^ns. 
C = 58° 46'.9 Ans. 

(Check.) 



1.91589 
2.24586 
9.67003-10 

1.91589 
2.06198 
9.85391 - 10 

1.91589 
2.16518 
9.75071-10 



179° 59\9 

*By adding s - 



(§37) 



-«, 5—6, 5 — C. 



70 PLANE TRIGONOMETRY [VI, § 51 

EXERCISES 

Solve and check each of the following triangles : 

1. a = 2.4169, b = 3.2417, c = 4.6293. 

2. a = 21.687, b = 10.429, c = 14.221. 
Z. a = 528.62, b = 499.82, c = 321.77. 

4. a = 2179.1, 6 = 3467.0, = 5061.8. 

5. a = 0.1214, 6 = 0.0961, c = 0.1573. 

6. Eind the areas of the triangles in Exs. 1-5. 

7. Find the areas of the inscribed circles of the triangles in Exs. 1-5. 

OTHER LOGARITHMIC COMPUTATIONS 
52. Interest and Annuities. 

Simple Interest. 

Let the principal be represented by F 

the interest on $ 1 for one year by r 

the number of years by n 

the amount of P for n years by A^ 

Then the simple interest on P for a year is Pr 

the amount of P for a year is P -\-Pr =P(l+r), 

the simple interest on P for n years is Pur 

the amount of P for n years is A^^ = P(l + nr). 

Example. How long will it take $5 210, at 4% simple interest, to 
amount to ^ 298.20 ? 

An = P(l + nr) i.e. n = ^" ~ ^ . 

Pr 

Number Logarithm 

An- P= 88.20 -^ 1.9455 

Pr= 8.40 -> 0.9243 

n = 10.5 <- 1.0212 10 yr. 6 mo. ^ns. 

Compound Interest. 
Let the original principal be P 

and the rate of interest r 

Then the amount A^ at the end of the first year is 
A, = P-{-Pr=P(l + r), 



VI, § 52] LOGARITHMIC COMPUTATION 71 , 

the amount A2 at the end of the second year is 

A, = A,{1 + r) = P(l + r)^ 
the amount at the end of n years is 

A^ = Pil + ry. 
If the interest is compounded semiannually, ^„ — P[ 1 + - 



if quarterly ^„ = Pf 1 + - ) , if g times a year A,^ = P( 1 -^- 
\ V _ \ 9J 

Since P in n years will amount to A,^, it is evident that P at 

the present time may be considered as equivalent in value to 

A due at the end of ii years. Hence P is called the present 

worth of a given future sum A. Since 

A„ = P(l + ry, P=A„il + r)-''. 

Example. In how many years will one dollar double itself at 4 9^ in- 
terest compounded annually ? 

An = P(l + r)^ or log — " = nlog(l + r). 

. ^^ logA-lQg^ 
log (1 + r) 

Hence ^ ^log2 - logl ^ OgOlO ^ 1,,^^ 

log (1.04) 0.0170 

17 yr. 9 mo. Ans. 

AxxuiTiES. An annuity is a fixed sum of money payable 
at equal intervals of time. 

To find the present worth of an annuity of A dollars pay- 
able annually for n years, beginning one year hence, the rate 
of interest being r and the number of years n. 

Since the present worth of the first payment is A(l + r)~-, 
of the second ^(1 + r)~^, etc., the present worth of the whole 

P=^[(l + r)-i+(l + r)-2+ ... +(l + r)-]. 

The quantity in the brackets is a G. P. whose ratio is (1 + r)~K 
Summing, we have 

l-(l+r)-i 



(1 + ryj 



72 PLANE TRIGONOMETRY [VI, § 52 

If the annuity is perpetual, i.e. n is infinite, the formula for 
present worth becomes P — —. 

Example. What should be paid for an annuity of 1 100 payable an- 
nually for 20 years, money being worth 4 % per annum ? 

p=mi L__]. 

0.04 L (1.04)2oJ 
By logarithms ( 1 .04) 20 = 2. 188. 

Therefore P= — [l ^1 =2500 [ lil??1 =$1358, approximately. 

0.04 L 2.188 J L2.I88J ^ 

53. Projectiles. Logarithms are used extensively in ballis- 
tic computations. [Ballistics is the science of the motion of 
a projectile.] The following is a very simple example of the 
type of problem considered. 

The time of flight of a projectile (in vacuum) is given by 

the formula T=^^ ^ where X is the horizontal range 

in feet, <^ is the angle of departure, and g is the acceleration 
due to gravity in feet per second per second [g = 32.2]. If it 
is known that the range is 3000 yd. and that the angle of de- 
parture is 30° 20', find the time of flight. 



T = 4 



2 X tan </) 
9 



Numbers Logarithms 

2 X= 18000 -> 4.2553 
tan 30° 20' -> 9.7673 - 10 

4.0226 
32.2 -> 1.5079 

2)2.5147 
18.09 <- 1.2574 r = 18.09 seconds. Ans. 

EXERCISES 

1. Find the amount of $ 500 in 10 years at 4 per cent compound inter- 
est, compounded semiannually. 

2. In how many years will a sum of money double itself at 5 per cent 
interest compounded annually ? semiannually ? 



VI, § 54] LOGARITHMIC COMPUTATION 73' 

3. A therinoinotor bulb at a temperature of 20"^ C. is exposed to the air 
for 15 seconds, in wliich time the temperature drops 4 degrees. If the 
hiw of cooUng is giveu by the formuhi = ^oe~''S where 6 is the tinal tem- 
perature, ^0 the initial temperature, e the natural base of logarithms, and, 
t the time in seconds, find the value of b. 

4. The stretch s of a brass wire when a weight m is hung at its free 
end is given by the formula ^^^ j 

where m is the weight applied in grams, g = 980, / is the length of the 
wire in centimeters, r is the radius of the wire in centimeters, and A: is a 
constant. If m = 844.9 grams, I = 200.9 centimeters, r = 0.30 centi- 
meter when s = 0.056, find k. 

5. The crushing weight P in pounds of a wrought-iron column is given 

by the formula .73.-5 

P = 299,600^^ — , 

where d is the diameter in inches and I is the length in feet. What weight 
will crush a \NTOught-iron column 10 feet long and 2.7 inches in diameter? 

6. The number n of vibrations per second made by a stretched string 
is given by the relation -1 -^p- 

2i\^' 

where I is the length of the string in centimeters, 3/ is the weight in 
grams that stretches the string, m the weight in grams of one centimeter 
of the string, and g = 980. Find n when M = 5467.9 grams, i = 78.5 
centimeters, m = 0.0065 gram. 

7. The time t of oscillation of a pendulum of length I centimeters is 
given by the formula — -— 

>980 
Find the time of oscillation of a pendulum 73.27 centimeters in length. 

8. The weight w in grams of a cubic meter of aqueous vapor saturated 
at 17^ C. is given by the formula 

^f. ^ 1203 X 12.7 X 5 

(l + iV3)(760x8)' 
Compute IV. 

54. The Logarithmic Scale. An arithmetic scale in which the 
segments from the origin are proportional to the logarithms of 1, 2, 3, etc., 
is called a logarithmic scale. Such a scale is given in Fig. 42. 

I I I 1 — I I I I I 1 

1 f 3 4 5 6 7 8 9 10 

Fig. 42 



4 



PLANE TRIGONOMETRY 



[VI, § 



55. The Slide Rule. The slide rule consists of a rule along the 
center of which a slip of the same material slides in a groove. Along the 





.1 


2 

1 1 


1 '.T'l 


5 


6 


7 8 9 1 


1 


! 


^^ 


3 




^ 


5 


6 7 89 








A ^v^ 
































R r i ' ' ' 






































■' 37r ^ 


s 


R 






2 




3 




^ 


5 


e - e 9 
























-^ 










C ^T-r- 
























'■'Hi 






n i' i •!- 
























lli|ln 




1 






1 




2 






3 


4 




5 




6 




(7 


8 9 





Fig. 43 

upper edge of the groove are engraved two logarithmic scales, A and J5, , 
that are identical. Along the low^er edge are also two identical logarithmic 
scales, G and D, in which the unit is twice that in scales A and B. Since 
the segments represent the logarithms of the numbers found in the scale, 
the operation of adding the segments is equivalent to multiplying the 



r 

1 2 




3^ 

i I 


4 

1 


5 


6 7 8 9 1 

! III! 




2 

1 I 


A 1 ! 1 I M M ' 1 1 ' ' ■ ' ' ' i ' ''MM 


^1 1 hj Ml 1 III I I lilllllii: ' . ^' I'lllilil 


\ ( 


D ' — 












i : ''lilli 


\ 




^r 




1 


2 


' Stt 4 


5 


6 7 8 9 


I 


c^- 








! , , , 2 




1 , , . , ? 


/ 


V 












1 1 ' 1 


mU 


T 


'7 


i \ 


^l|||||llllllllllllll|lilil!l' . • , . . r '--.i -r-!:Mi||!l|mi 


Nil 


1 


1 1 


|i 










1 


1 




1 
2 




-co 


1 


T ' 



Fig. 44 

corresponding numbers. Thus in Fig. 44 the point marked 1 on scale B 
is set opposite the point marked 2.5 on scale A. The point marked 4 on 
scale B will be opposite the point marked 10 on scale A, i.e. 2.6 x 4: = 10. 
Similarly we read 2.5 x 3.2 = 8, 2.5 x 2.5 = 6.25. Other multiphcations 
can be performed in an analogous manner. 

Division can be performed by reversing the operation. Thus in Fig. 44 
every number of scale B is the result of dividing the number above it by 
2.5. Thus we read 7.2-4- 2.5 = 2.9 approximately. 

Since scales C and D are twice as large as scales A and B, it follows 
that the numbers in these scales are the square roots of the numbers 
opposite to them in scales A and B. Conversely the numbers on scales 
A and B are the squares of the numbers opposite them on scales C and 
D. Moreover the scales C and D can be used for multiplying and divid- 
ing, but the range of numbers is not so large. • 

For a more complete discussion of the use of a sHde rule consult the 
book of instructions published by any of the manufacturers of slide rules, 
where also exercises will be found for practice. 



CHAPTER VII 
TRIGONOMETRIC RELATIONS 

56. Radian Measure. In certain kinds of work it is more 
convenient in measuring angles to use, instead of the degree, 
a unit called the radian. A radian is defined as the angle at 
the center of a circle whose subtended arc is equal in length 
to the radius of the circle (Fig. 45). Therefore, if an angle 
at the center of a circle of radius 7' units subtends an arc of 
s units, the measure of in radians is 

(1) 6 = ^. 

Since the length of the whole circle is 2 7rr, it follows that 

— = 2 TT radians = 360^ 
/• 

or 

(2) IT radians = 180°. 
Therefore, 

180° _ _ 

1 radian = = o<° 1/ 45" (approximately). p^^ ^- 

TT 

It is important to note that the radian ^ as delined is a con- 
stant angle, i.e. it is the same for all circles, and can therefore 
be used as a unit of measure. 

From relation (2) it follows that to convert radians into 
degrees it is only necessary to multiply the number of radians 
by ISO/tt, while to convert degrees into radians we multiply 
the number of degrees by 7r/180. Thus 45° is tt '4 radians ; 
TT 2 radians is 90°. 

* The symbol *" is ofteif used to denote radians. Thus 2'' stands for 2 
radians, tt'' for tt radians, etc. When the angle is expressed in terms of tt (the 
radian being the unit), it is customary to omit ^ Thus, when we refer to an 
angle tt, we mean an angle of tt radians. AVhen the word radian is omitted, 
it should be mentally supplied in order to avoid the error of supposing tt 
means 180. Here, as in geometry, w = .*>. 14159. . . . 





76 PLANE TRIGONOMETRY \yil, § 57 

57. The Length of Arc of a Circle. Erom relation (1), 

§ 56, it follows that 

That is (Fig. 46), if a central angle is measured 
in radians, and if its intercepted arc and the 
radius of the circle are measured in terms of 
the same unit, then 
length of arc = radius X central angle in radians. 

EXERCISES 

1. Express the following angles in radians : 

25°, 145°, 225°, 300°, 270°, 450°, 1150°. 

2. Express in degrees the following angles : 

TT TtT StT q 5 it 

4' Q' Q' '4 

3. A circle has a radius of 20 inches. How many radians are there in 
an angle at the center subtended by an arc of 25 inches ? How many 
degrees are there in this same angle ? Arts, j*" ; 71° 37' approx. 

4. Find the radius of a circle in which an arc 12 inches long subtends 
an angle of 35°. 

5. The minute hand of a clock is 4 feet long. How far does its ex- 
tremity move in 22 minutes ? 

6. In how many hours is a point on the equator carried by the rotation 
of the earth on its axis through a distance equal to the diameter of the earth? 

7. A train is traveling at the rate of 10 miles per hour on a curve of 
half a mile radius. Through what angle has it turned in one minute ? 

8. A wheel 10 inches in diameter is belted to a wheel 3 inches in 
diameter. If the first wheel rotates at the rate of 5 revolutions per 
minute, at what rate is the second rotating ? How fast must the former 
rotate in order to produce 6000 revolutions per minute in the latter ? 

58. Angular Measurement in Artillery Service. The 

divided circles by means of which the guns of the United States Field 
Artillery are aimed are graduated neither in degrees nor in radians, but 
in units called mils. The mil is defined as an angle subtended by an arc 
of 6^/00 ^^ ^^^^ circumference, and is therefore equal to 
2 7r 3.1416 



6400 3200 



= 0.00098175 =(0.001 - 0.00001825) radian. 



VII, § 58] TRIGONOMETRIC RELATIONS 



77 



The mil is therefore approximately one thousandth of a radian. 
(Hence its name.)* 

Since (§57) 
length of arc = radius x central angle in radians, 
it follows that we have approximately 



lenixth of arc : 



radius 
■ lOOO' 



X central angle in mils ; 



(angle 



i.e. length of arc in yards = (radius in thousands of yards) 
in mils). The error here is about 2 ^c. 

Example 1. A battery occupies a front of 60 yd. If it is 
at 5500 yd. rans^e, what an^jle does it subtend (Fisr. 47)? We 
have, evidently, j^j^ ^j 

anffle = — = 11 mils. 
^ 5.5 

Example 2. Indirect Fire, f A battery 
posted with its right gun at G is to open fire on 
a battery at a point T, distant 2000 yd. and in- 
visible from G (Eig. 48) . The officer directing 
the fire takes post at a point B from which both 
the target T and a church spire P, distant 
3000 yd. from G, are visible. B is 100 yd. at 
the right of the Une G T and 120 yd. at the 
right of the line GP and the officer finds by 
measurement that the angle PBT contains 
3145 mils. In order to train the gun on the 
target the gunner must set off the angle PG T 
on the sight of the piece and then move the gun 

* To give an idea of the value in mils of certain angles the following has 
been taken from the Drill Regulations for Field Artillery (1911), p. 164: 

" Hold the hand vertically, palm outward, arm fully extended to the front. 
Then the angle subtended by the 

width of thumb is 40 mils 

width of first finger at second joint is 40 mils 

width of second finger at second joint is .... 40 mils 

width of third finger at second joint is 35 mils 

width of little finger at second joint is 30 mils 

width of first, second, and third fingers at second joint is . 115 mils 
These are average values." 

t The limits of the text preclude giving more than a single illustration of 
the problems arising in artillery practice. For other problems the student is 
referred to the Drill Regulations for Field Artillery (1911), pp. 57, 61, 150-164 ; 
and to Andrews, Fundamental of Military Service, pp. 153-159, from which 
latter text the above example is taken. 




78 PLANE TRIGONOMETRY [VII, § 58 

until the spire P is visible through the sight. When this is effected, the 
gun is aimed at T. 

Let F and E be the feet of the perpendiculars from B to GT and GP 
respectively, and let BT' and BP' be the parallels to GT and GP that 
pass through B. Then, evidently, if the officer at B measures the angle 
PB r, which would be used instead of angle PG T were the gun at B in- 
stead of at G, and determines the angles TBT' = FTB and PBP' = EPB, 
he can find the angle PG T from the relation 

PGT = P'BT' = PBT- TBT' - PBPf. 

Now tan FTB = ^,td.n EPB = ^ • 

TF PE 

Furthermore if FTB and EPB are small angles, i.e. if FB and EB are 
small compared with G T and GP respectively, the radian measure of the 
angle is approximately equal to the tangent of the angle. Why ? Hence 
we have 

FTB = t^n FTB =z^] 

G T \ 

\ approximately. 

EPB = tan EPB = ^^ \ 

GP] 

Therefore TBT = FTB = J^ radians = 50 mils, 

2000 

PBP' = EPB = ^^ radians = 40 mils. 
3000 

Hence PGT = PBT - TBV - PBP' 

- 3145 - 50 - 40 

= 3055 mils, 
which is the angle to be set off on the sight of the gun. 

Hence from the situation indicated in Fig. 48 we have the following 
rule : 

(1) Measure in mils the angle PBT from the aiming point P to the 
target T as seen at B. 

(2) Measure or estimate the offsets FB and EB in yards, the range 
GT and the distance GP of the aiming point P in thousands of yards. 

(3) Compute in mils the offset angles by means of the relations 

TBT = FTB, 
PBP' = EPB, 

TBT' = ^. 
GT 

PBP' = — . 
GP 

(4) Then the angle of deflection PGT is equal to the angle PBT 
diminished by the sum of the offset angles. 



VII, § 59] TRIGONOMETRIC RELATIONS 79 



EXERCISES 

1. A battery occupies a front of 80 yd. It is at 5000 yd. range. 
What angle does it subtend ? 

2. In Fig. 48 suppose PBT = 3000 mils, FB = 200 yd., GT = 3000 yd., 
EB = 150 yd., GP= 4000 yd. Find the number of mils in PGT. 

3. A battery at a point G is ordered to take a masked position and be 
ready to fire on an indicated hostile battery at a point T whose range is 
known to be 2100 yd. The battery commander finds an observing station 
5, 200 yd. at the right and on the prolongation of the battery front, and 
175 yd. at the right of PG. An aiming point P, 5900 yd. in the rear, is 
found, and PB T is found to be 2600 mils. Find PG T. 

4. A battery at a point G is to fire on an invisible object at a point T 
whose range is known to be 2000 yd. A battery commander finds an 
observing station B, 100 yd. at the right of GT and 150 yd. at the right 
of GP. The aiming point P is 1500 yd. in front and to the left of GT. 
The angle TBP contains 1200 mils. Find PG T. 



59. The Sine Function. Let us trace in a general way the 
variation of the function sin ^ as ^ increases from 0° to 360°. 
For this purpose it will be convenient to think of the distance 
r as constant, from which it follows that 
the locus of P is a circle. When 6 = 0°, the 
point P lies on the o^-axis and hence the 
ordinate is 0, i.e. sin 0° = 0/n =0. As ^ 
increases to 90°, the ordinate increases 
until 90° is reached, when it becomes equal 
to r. Therefore, sin 90° = r/r = 1. As ^ Pig. 49 

increases from 90° to 180°, the ordinate de- 
creases until 180° is reached, when it becomes 0. Therefore 
sin 180° = 0/r = As ^ increases from 180° to 270°, the ordi- 
nate of P continually decreases algebraically and reaches its 
smallest algebraic value when 6 = 270°. In this position the 
ordinate is — r and sin 270° = — r/r = — 1. When enters 
the fourth quadrant, the ordinate of P increases (algebraically) 
until the angle reaches 360°, when the ordinate becomes 0. 




80 



PLANE TRIGONOMETRY 



[VII, § 59 



Hence, sin 360° = 0. It then appears that : 

as 6 increases from 0° to 90°, sin 6 increases from to 1 ; 

as 6 increases from 90° to 180°, sin 6 decreases from 1 to ; 

as 6 increases from 180° to 270°, sin 6 decreases from to — 1 ; 

as 6 increases from 270° to 360°, sin 6 increases from — 1 to 0. 
It is evident that the function sin 6 repeats its values in the 
same order no matter how many times the point P moves 
around the circle. We express this fact by saying that the 
function sin 6 is periodic and has a period of 360°. In symbols 
this is expressed by the equation 

sin [0 + n . 360°] = sin 9, 

where n is any positive or negative integer. 

The variation of the function sin 6 is well shown by its 
graph. To construct this graph proceed as follows : Take a 
system of rectangular axes and construct a circle of unit radius 

T 




Fig. 50 



with its center on the ir-axis (Fig. 50). Let angle XM^P = 6, 
Then the values of sin 6 for certain values of are shown in 
the unit circle as the ordinates of the end of the radius drawn 
at an angle 0. 



e 





30^ 


45^^ 


60° 


90° 




sin^ 





3fiPi 


M,P2 


M^P, 


M,P, 


... 



Now let the number of degrees in 6 be represented by dis- 
tances measured along OX. At a distance that represents 30° 
erect a perpendicular equal in length to sin 30° ; at a distance 



VII, § 60] TRIGONOMETRIC RELATIONS 



81 



that represents 60° erect one equal in length to sin 60°, etc. 
Through the points 0, Pi, jPo? •*• draw a smooth curve ; this 
curve is the graph of the function sin 6, 

If from any point P on this graph a perpendicular PQ is 
drawn to the ;r-axis, then QP represents the sine of the angle 
represented by the segment OQ. 

Since the function is periodic, the complete graph extends 
indefinitely in both directions from the origin (Fig. 51). 



T 

1 




/'\ 


/\ /\ 


' \y 


lbO\ /360^ 5iO^X 


-1 


y = sin X 



Fig. 51 

60. The Cosine Function. By arguments similar to those 
used in the case of the sine function we may show that : 
as 6 increases from 0° to 90°, the cos decreases from 1 to ; 
as 6 increases from 90° to 180°, the cos decreases from to — 1 ; 
as 9 increases from 180° to 270°, the cos increases from — 1 to ; 
as increases from 270° to 360°, the cos increases from to 1. 

The graph of the function is readily constructed by a method 




Fig. 52 



similar to that used in the case of the sine function. This is 
illustrated in Fig. 52. 

The complete graph of the cosine function, like that of the 
sine function, will extend indefinitely from the origin in both 



82 



PLANE TRIGONOMETRY 



[VII, § 60 



directions (Fig. 53). Moreover cos^, like sin^, i'^ periodic and 
has a period of 360°, i.e. 

cos [9 + n-360°]= cos 9, 

where n is any positive or negative integer. 



7 

1 




^^ /^ 


A /^^ 


Vy ° 


\/ \'^ 


-1 


y=t'Osa: 



Fig. 53 




61. The Tangent Function. In order to trace the varia- 
tion of the tangent function, consider a circle of unit radius 
with its center at the origin of a system of rectangular axes 
(Fig. 54). Then construct the tangent to 
this circle at the point M(l, 0) and let P 
denote any point on this tangent line. If 
angle MOP= 0, we have tan 6 = MP/OM 
= MP /I = MP, i.e. the line MP represents 
tan 6. 

Now when 6 = 0°, MP is 0, i.e. tan 0° is 0. 
As the angle increases, tan 6 increases. As 
approaches 90° as a limit, 3IP becomes 
infinite, i.e. tan 6 becomes larger than any number whatever. 

At 90° the tangent is undefined. It is sometimes convenient 
to express this fact by writing 

tan 90° =00. 

However we must remember that this is not a definition for 
tan 90°, for oo is not a number. This is merely a short way of 
saying that as approaches 90° tan becomes infinite and 
that at 90° tan is undefined. 

Thus far we have assumed 6 to be an acute angle approach- 
ing 90° as a limit. Now let us start with as an obtuse angle 



Fig. 54 



VII, § on 



TRIGONOMETRIC RI^.ATIONS 



83 




and let it decrease towards 90° as a limit. In Fig. ryB the line 
MP' (which is here negative in direction) represents tan 6, 
Arguing precisely as we did before, it is 
seen that as the angle 6 aj)proaches 90° 
as a limit, tan again increases in magni- 
tude beyond all bounds, i,e. becomes inh- 
nite, remaining, however, always negative. 
We then have the following results. 

(1) When 6 is acute and increases to- 
wards 90° as a limit, tan always remains 
positive but becomes infinite. At 90° tan is undefined. 

(2) When 9 is obtuse and decreases towards 90° as a limit, 
tan always remains negative but becomes infinite. At 90° 
tan is undefined. 

It is left as an exercise to finish tracing the variation of the 
tangent function as varies from 90° to 360°. Note that 
tan 270°, like tan 90°, is undefined. In fact tan n • 90° is unde- 
fined, if n is any odd integer. 



Fig. 




Fig. 5G 



To construct the graph of the function tan we proceed 
along lines similar to those used in constructing the graph of 
sin and cos 0. The following table together with Fig. 50 
illustrates the method. 



84 



PLANE TRIGONOMETRY 



[\1I, § 61 



e 


0° 



30° 


45° 


60° 


90° 


120° 


135° 


150° 


180° 


210° 


tan^ 


MPi 


MP2 


MP^ 


undefined 


MP^ 


MP^ 


MP^ 


JfP7 = 


MPi 



It is important to notice that tan 6, like sin 6 and cos d^ is 
periodic, but its period is 180°. That is 

tan(e + n • 180°)= tan e, 

where n is any positive or negative integer. 

EXERCISES 

1. What is meant by the period of a trigonometric function ? 

2. What is the period of sin d ? cos 6 ? tan d ? 

3. Is sin 6 defined for all angles ? cos d ? 

4. Explain why tan 6 is undefined for certain angles. Kame four 
angles for which it is undefined. Are there any others ? 

5. Is sin (<9 + 360°) = sin 6 ? 

6. Is sin (^+ 180°) = sin 6*? 

7. Is tan (^ + 180°) = tan d ? 

8. Is tan (0 + 360°) = tan^ ? 

Draw the graphs of the following functions and explain how from the 
graph you can tell the period of the function : 

9. sin^. 11. tan^. 13. sec ^. 
10. cos (9. 12. csc^. 14. ctn6>. 

Verify the following statements : 

15. sin90° + sin270° = 0. 18. cos 180° + sin 180° =- 1. 

16. cos90° + sin0° = 0. 19. tan 360° + cos 360° = 1. 

17. tanl80° + cosl80°=- 1. 20. cos 90° + tan 180°- sin 270° = !.' 

21. Draw the graphs of the functions sin 6, cos 6, tan 6, making use of 
a table of natural functions. See p. 112. 

22. Draw the curves y = 2smd ; y = 2cose ; y = 2 tan d. 

23. Draw the curve y = sin 6 + cos 6. 

24. From the graphs determine values of 6 for which sin 6 = ^ ; sin ^ 
= 1 ; tan ^ = 1 ; cos ^ = J ; cos ^ = 1. 



VII, § 63] TRIGONOMETRIC RELATIONS 



85 



62. The Trigonometric Functions of — 9. Draw the angles 
6 and — 6, where OP is the terminal line of 6 and OP is the 
terminal line of — 6. Figure 57 shows an angle 6 in each of 




Fig 57 



the four quadrants. We shall choose OP = OP and (.r, y) as 
the coordinates of P and (x\ y') as the coordinates of P', In 
all four figures 

x' = x, 2/' = - y, r' = r. 
Hence 

, ^/' -. — V. 



sin (-6')^ 



: — sin ^, 



cos(-6>) 



— = - = cos 0^ 
v' r 



tan (-6*)--^ 



■ = — tan 0, 



Also, 

csc(— ^)=— csc(9 ; sec(— ^)= sec^ ; ctn (— ^)= — ctn^. 
The above results can be stated as follows : The functions of 
— 6 equal numerically the like named functions of 0. The 
algebraic sign, however, will be opposite except for the cosine 
and secant. 

Example. sin-10^=— sinlO'', cos— 10^ = cos10'', tan— 10^= -tan 10°. 

63. The Trigonometric Functions of 180° + 0. Similarly, 
the following relations hold : 

sin (180° + (9) = - sin 6, esc (180° + ^) = - esc (9, 

cos (180° + (9) = - cos e, sec (180° -{-0)=- sec (9, 

tan (180° + (9) = tan (9, ctn (180° -h 0) = ctn 0. 

The proof is left as an exercise. 



86 PLANE TRIGONOMETRY [VII, § 64 

64. Summary. An inspection of the results of §§ 27-28, 
62-63 shows : 

1. Each function of — or 180"^ ± 6 is equal in absolute value 
(but not always in sign) to the same function of 0. 

2. Each function of 90° — is equal in magnitude and in sign 
to the corresponding co function of 6. 

These principles enable us to find the value of any function 
of any angle in terms of a function of a positive acute angle 
(not greater than 45° if desired) as the following examples 
show. 

Example 1. Beduce cos 200° to a function of an angle less than 45°. 

Since 200° is in the third quadrant, cos 200° is negative. Hence 
cos 200° = - cos 20°. Why ? 

Example 2. Reduce tan 260° to a function of an angle less than 45°. 

Since 260° is in the third quadrant, tan 260° is positive. Hence 
tan 260° = tan 80° = ctn 10° (§ 27). 

Example 3. Reduce sin (— 210°) to a function of a positive angle 
less than 45°. 

From § 62 we know sin — 210° = — sin 210°. 

Considering the positive angle 210°, we have 

sin - 210° = - sin 210° = - [ - sin 30°] = sin 30°. 

EXERCISES 

Reduce to a function of an angle not greater than 45° : 

1. sin 163°. 5. esc 901°. 

2. cos(-110°). 6. ctn(~1215°). 

Ans, -sin 20°. 7. tan 840°. 

3. sec (-265°). 8. sin 510°. 

4. tan 428°. 

Find without the use of tables the values of the following functions : 
9. cos 570°. 11. tan 390°. 13. cos 150°. 

10. sin 330°. 12. sin 420°. 14. tan 300°. 

Reduce the following to functions of positive acute angles : 

15. sin 250°. 18. sec (-245°). 

Ans. — sin 70° or — cos 20°. 19. csc(— 321°). 

16. cos 158°. 20. sin 269°. 

17. tan (-389°). 



VII, § 64] TRIGONOMETRIC RELATIONS 87 

Prove the following relations from a figure : 



(«) sin (90" + ^)=cos^. 




(c) 


sin (180° + ^) = -sin6>. 


cos (90'^ + e) = - sin d. 






cos (180° +6) = — cos^'. 


tan (90^ + 0) = - ctn d. 






tan (180° + 0)= tan (9. 


csc(90° + 6>)=sec^. 






CSC(180° + ^):=- csc^. 


sec (90^ + d) = - CSC e. 






sec (180° + 6) = -sec 6. 


ctn (90^ -{-$) = — tan d. 






ctn (180° + 0)=z ctn^. 


(b) sin (180'='- 6/)=sin^. 




(d) 


sin (270° -(9) = — cos (9. 


cos (180^-^) = -cos (9. 






cos (270° — $) = - sin 6. 


tan (180° - 6') = -tan6>. 






tan (270° — 6)= ctn 0. 


CSC (180^-^)= csc^. 






CSC (270°-^) = -sec (9. 


sec (180^- ^) = - sec^. 






sec (270°— 0) = -csc0. 


Ctn (180^-^) = - ctn^. 






ctn (270° -6)= tan 0. 


(e) sin (270° 


+ e): 


= — 


cos^. 


cos (270° 


+ 6): 


= sin 6. 


tan (270° 


+ e): 


= — 


ctn^. 


CSC (270° 


+ e): 


= - 


sec^. 


sec (270° 


+ e): 


= CSC 6. 



ctn (270° + ^) = — tan^. 



CHAPTER YIII 

TRIGONOMETRIC RELATIONS (Continued) 

65. Trigonometric Equations. An identity, as we have 
seen (§ 26), is an equality between two expressions which is 
satisfied for all values of the variables for which both expres- 
sions are defined. If the equality is not satisfied for all 
values of the variables for which each side is defined, it is 
called a conditional equality, or simply an equation. Thus 
1 — cos ^ = is true only ii = n - 360°, where n is an integer. 
To solve a trigonometric equation, i.e. to find the values of 
for which the equality is true, we usually proceed as follows. 

1. Express all the trigonometric functions involved in terms 
of one trigonometric function of the sa^ne angle. 

2. Find the value (or values) of this function by ordinary 
algebraic methods. 

3. Find the angles between 0° and 360° which correspond to 
the values found. These angles are called particular solutions. 

4. Give the general solution by adding n • 360°, where n is 
any integer, to the particular solutions. 

Example 1. Find 6 when sin 6 = I. 

The particular solutions are 30° and 150*^. The general solutions are 
30° + n . 360°, 150° + n - 360°. 

Example 2. Solve the equation tan ^ sin ^ — sin ^ = 0. 

Factoring the expression, we have sin ^(tan ^ — 1) = 0. Hence we 
have sin ^ = 0, or tan ^—1 = 0. Why ? 

The particular solutions are therefore 0°, 180°, 45°, 225°. The general 
solutions are n • 360°, 180° -f n • 360^ 45° + ^ • 360°, 225° + n • 360°. 

88 



1. 


2 


2. 


2 ^ 


3. 


2 


4. 


2 


5. 


tan^ =—1. 


6. 


ctn<9 = 1. 


16. 


2 sin ^ = tan 0, 



VIII, § 66] TRIGONOMETRIC RELATIONS 89 

EXERCISES 

Give the particular and tlie general solutions of the following 
equations : 

V3 . 7. sec ^ = 2. 

8. tan ^ = 0. 

9. sec2(? = 2. 

10. sin2 6 = 1. 

11. cos^=— J. 

12. csc2 d = f . 

13. 4 sin ^ — 3 CSC ^ = 0. 

14. 2sin^cos2^ = sin^. 

15. cos $ -f sec ^ = f . 
Ans. Particular solutions : 0", 180°, 60°, 300°. 

17. 3 sin ^ + 2 cos ^ = 2. 18. 2 cos2 ^ — 1 = 1 _ sin2 6. 

66. Inverse Trigonometric Functions. The equation 

X = sin y (1) 

may be read : 

y is an angle whose sine is equal to x, 

a statement which is usually written in the contracted form 

y = arc sin x* (2) 

For example, x = sin 30° means that x = ^, wliile y = arc sin ^ 

means that y = 30°, 150°, or in general (n being an integer), 

30° + 71 ' 360° ; 150° + n • 360°. 

Since the sine is never greater than 1 and never less than 

— 1, it follows that — 1 ^ x ^ 1. It is evident that there is 

an unlimited number of values oi y = arc sin x for a given value 

of X in this interval. 

We shall now define the principal value Arc sin a; f of arc sin x, 

distinguished from arc sin x by the use of the capital A, to be 

* Sometimes written y = sin -i x. Here — 1 is not an algebraic exponent, 
but merely a part of a functional symbol. When we wish to raise sin x to 
the power — 1, we write (sin x)-^. 

t Sometimes written Sin-i x, distinguished from sin-i x by the use of the 
capital S. 



90 



PLANE TRIGONOMETRY 



[VIII, § 66 



the numerically smallest angle whose sine is equal to x. This func- 
tion like arc sin a? is defined only for those values of x for 
which 

The difference between arc sin a; and Arc sin a? is well illus- 
trated by means of their graph. It is 
evident that the graph oi y = arc sin x, 
i.e, X = sin y is simply the sine curve 
with the role of the x and y axes inter- 
changed. (See Fig. 58.) Then for every 
admissible value of x, there is an un- 
limited number of values of y ; namely, 
the ordinates of all the points Pi, Po, •••, in 
which a line at a distance x and parallel 
to the ?/-axis intersects the curve. The 
single-valued function Arc sin x is repre- 
sented by the part of the graph between 
IT and N. 
Similarly arc cos x, defined as " an angle whose cosine is x/^ 

has an unlimited number of values for 

every admissible value ofa;(— l^x<l) 

We shall define the principal value Arc 

cos X as the smallest positive angle whose 

cosine is x. That is, 

^ Arc cos x < IT. 



Eigure 59 represents the graph oi y = arc 
cos X, and the portion of this graph between 
M and A^ represents Arc cos x. 

Similarly we write x = tan y 2is y = arc 
tan Xy and in the same way we define the 







T 












Sir 




% 








sir 

2 












TT 




\ 








TT 

2 


^ 


> 

y 


K 


-1 


y 


^ 






Tx' 


M 


( 


IT 

2 














y^ arc sin x 








y= 


=j.> 


c sin X 



Fig. 58 





Y 










^TT 




) 






STT 




i 




iV 


(: 










TT 

2 


■v. 


\ 


M 


-1 




IT 






1 X 



y= arc cos x 
y=Arc cos X 
Fig. 59 



symbols arc ctnoj; arc seca^; arc esc a;. 

The principal values of all the inverse trigonometric functions 

are given in the following table. 



VIII, § 66] TRIGONOMETRIC RELATIONS 



91 



y = 


Arc sin x 


Arc cosx 


Arc tan x 


Range of x 


-l^x^ 1 


-l^x^l 


all real values 


Range of y 


-!Lto^ 
2 2 


to TT 


to — 

2 2 


X positive 
X negative 


1st Quad. 
4th Quad. 


1st Quad. 
2d Quad. 


1st Quad. 
4th Quad. 




Arc ctnx 


Arc sec x 


Arc CSC X 


Range of x 


all values 


x^lorx^— 1 


x^l orx^"— 1 


Range of y 


to TT 


to TT 


to - 

2 2 


X positive 
X negative 


1st Quad. 
2d Quad. 


1st Quad. 
2d Quad. 


1st Quad. 
4th Quad. 



In so far as is possible we select the principal value of each 
inverse function, and its range, so that the function is single- 
valued, continuous, and takes on all possible values. This ob- 
viously cannot be done for the Arc sec x and for Arc esc y. 



(/) Arc cos -^. 



EXERCISES 

1. Explain the difference between arc sin x and Arc sin x. 

2. Find the values of the follow^ing expressions : 

(a) Arc sin J. (d) Arc tan — 1. 

(b) arc sin J. . . V3 
^ ^ ^ (e) arc cos -^. 

(c) arc tan 1. 2 

3. What is meant by the angle tt ? 7r/4 ? 

4. Through how many radians does the minute hand of a watch turn 
in 30 minutes ? in one hour ? in one and one half hours ? 

6. For what values of x are the following functions defined : 

(a) arc sin x ? (c) arc tan x ? (e) arc sec x ? 

(b) arc cos x ? (d) arc ctn x ? (/) arc esc x ? 

6. What is the range of values of the functions : 
(a) Arc sin x ? (c) Arc tan x ? (e) Arc sec x. 

(h) Arc cos X ? (<f) Arc ctn x ? (/) Arc esc x ? 



92 



PLANE TRIGONOMETRY 



[VIII, § 66 



(e) arc sec x, 
(/) arc CSC x. 



Then tan ^ = | and we wish to find the 



(e) sin (Arc sini). 
(/) tan (Arc tan 5) . 



7. Draw the graph of the functions : 
(a) arc sin x. (c) arc tan x. 
(h) arc cos x. (d) arc ctn x. 

8. Find the value of cos (Arc tan |) . 
Hint. Let Arc tan 

value of cos 6. 

9. Find the values of cos (arc tan |) . 

10. Find the value of the following expressions 
(a) sin (arc cos I). (c) cos (Arc cos j\). 
(5) sin (arc sec 3). (d) sec (Arc esc 2). 

11. Prove that Arc sin (2/5)=: Arc tan (2/V21) 

12. Find x when Arc cos (2 x2 — 2 x) = 2 7r/3. 
Find the values of the following expressions : 

13. cos [90° — Arc tan |]. • 

14. sec [OO*" — Arc sec 2]. 

15. tan[90°-- Arcsini-5_]. 

67. Projection. Consider two directed lines p and g in a 
plane, i.e, two lines on each of which one of the directions 
has been specified as positive (Fig. 60). Let A and B be 
any two points on p and let A', B' be the points in w^hich per- 

P. 




Fig. 60 



pendiculars to q through A and B, respectively, meet q. The 
directed segment A'B' is called the projection of the directed seg- 
ment AB on q and is denoted by 

AB' = proj^ AB. 
In both figures AB is positive. In the first figure A^B^ is posi- 
tive, while in the second figure it is negative. 

As special cases of this definition we note the following : 



VIII, § 67] TRIGONOMETRIC RELATIONS 93 

1. If }) and q are parallel and are directed in the same way, 

we have 

^m]^AB = AB. 

2. If p and q are parallel and are directed oppositely, we 

have 

proj^ AB = — AB, 

3. If 2^ is perpendicular to g, we have 

proj^ AB = 0. 
It should be noted carefully that these propositions arc true 

no matter how A and B are situated on p. 

We may now prove the following important proposition : 
If A and B are any two points on a directed line p, and q is 

any directed line in the same plane with p, then we have both 

in magnitude and sign 

(1) proj^ AB = AB ' cos (pq)^ = AB • cos (qp). 

We note first from § 8 that {pq)+ (qp)= + n • 360'', where 
n is any integer. . Hence from § 64, cos (pq) = cos (qp). Two 
cases arise. 



A' 




B' 



A qj Bi 

Fig. 61 

Case 1. Suppose AB is positive, i.e. it has the same direc- 
tion as p>. 

Through A draw a line q^ parallel to q and with the same 
direction. [It is evident that we may assume without loss of 
generality that q is horizontal and is directed to the right.] 
Let A^B^ be the projection of AB on q and let BB^ meet qi 
in ^1. Then by the definition of the cosine we have 

AB 

-—^ = cos (qip) — cos (pQ'i) = cos (gp) = cos {pq) 
AB 

* (P^) represents an angle through which p may be rotated in order to 
make its direction coincide with the direction of q ; similarly for {qp). 



94 PLANE TRIGONOMETRY [VIII, § 67 

in magnitude and sign. Hence 

ABi = AB • cos (pg) = AB ■ cos (qp). 
But AB, = A'B' = proj^ AB. 

Therefore proj^ AB = AB • cos (pp) = AB • cos (gjp). 

Case 2. Suppose AB is negative. 

If AB is negative, BA is positive and We have from Case 1, 
B'A' = BA • cos (pq)= BA • cos (qp). 
Changing the signs of both members of this equation, we have 
A^B' = AB • cos (pg)= AB • cos (qp). 

The special cases 1, 2, 3, are obtained from formula (1) 
by placing (qj:)) or (pq) equal to 0°, 180°, 90° respectively. 

Theorem. If A, B, C are any three points in a plane, a.nd I 
is any directed line in the p)lane, the algebraic sum of the projec- 
tions of the segments AB and BC on I is equal to the projection 
of the segment AC on I. 

As a point traces out the path from A to JB, and then from 
B to C (Eig. 62), the projection of the point traces out the 
segments from ^' to -B' and then from B^ 
to O. The net result of this motion is a 
motion from A^ to O which represents 
the projection of AC, i.e. 

AB^ + B'C = A'C\ 

EXERCISES 

1. What is the projection of a Hue segment upon a hne Z, if the hne 
segment is perpendicular to the hne I ? 

2. Find iproj^. AB and projy J.^^ in each of the following cases, if a 
denotes the angle from the x-axis to AB. 

(a) AB = 5, a = 60°. (c) AB = 6, a = 90°. 

(b) AB = 10, a = 300°. (d) AB = 20, a = 210°. 

* Projx ^B and projy AB mean the projections of AB on the a:-axis and 
the 2/-axis, respectively. 




VIII, § 68] TRIGONOMETRIC RELATIONS 



95 



3. Prove by means of projection that in a triangle ABC 

a = b cos C -\- c cos B. . 

4. If projj AB = 3 and proj^ AB =—4, find the length of AB. 

5. A steamer is going northeast 20 miles per hour. How fast is it 
going north ? going east ? 

6. A 20 lb. block is sliding down a 15° incline. Find what force 
acting directly up the plane will just hold the block, allowing one half a 
pound for friction. 

7. Prove that if the sides of a polygon are projected in order upon any 
given line, the sum of these projections is zero. 



r> 


P. 




•^ 












N^ 


/ 














■^"^ 


p 




xF 








~ 


y 







X' 




X 




r 


X 



68. The Addition Formulas. We may now derive formulas 
for sin (a + jS), cos (a + /3), and tan (a + /S) in terms of func- 
tions of a and /3. To this end 
let F (x, y) be any point on the 
terminal side of the angle a (the 
initial side being along the posi- 
tive end of the a^-axis and the 
vertex being at the origin). The 
angle a -{- (3 is then obtained by 
rotating OP through an angle 
)8. If P'{x', y') is the new posi- Fig. 68 

tion P after this rotation and 

OP = OP' = r, we have sin (« + m = ^, cos (a + i^) = - , by 

r r 

definition. Our first problem is, therefore, to find x^ and y' in 

terms of x, y, and /3. 

In the figure OJWP is the new position of the triangle OMP 
after rotating it about through the angle p. Now, 

x^ = proj^ OP' = proj, OJf ' + proj^ M'P 

= a;cos^ + 2/cos/'^ + |^ 

= x cos /3 — y sin ^. 



96 PLANE TRIGONOMETRY [VIII, § 68 

Similarly, 

y' = proj, OP' = proj, OM' + proj, M'P' 

(it ^ 

\2 ) 
= X sin P + y cos fi< 

Hence, • / , m v' ^ • o . V ^ 

' sm (a + ^) = -^ = - sm ^ 4- ^ cos B 

TV r 

or (1) sin (a + P) = sin a cos p + cos a sin p. 



Also 



cos (a + p) =~=z- cos /? — ^ sin /J. 
r r r 



or (2) cos (a H- p) = cos a cos p — sin a sin p. 

Further we have 

tan (a + 8)= ^^^ (^ + P) = sin a cos ^ + cos « sin ^8 , 
cos (a + y8) cos a cos ji — sin a sin ji 

Dividing numerator and denominator by cos a cos )8, we have 

(3) tan(a+B)= tana + tanp 
^ ^ V -r p; i_tanatanp 

Furthermore, by replacing yS by — ^ in (1), (2), and (3), and 
recalling that 

sin (— 13) =— sin jS, cos (— /3) = cos JS, tan (— y8) = — tan ^, 
we obtain 

(4) sin (a — P) = sin a cos p — cos a sin p, 

(5) cos (a — P) = cos a cos p + sin a sin p, 

(6) tan (a - p) = tana- tan p 
^ ^ ^ *^^ 1 + tan a tan p 

EXERCISES 

Expand the following : 

1. sin (45° -\-a)= 3. cos (60° + «) = 5. sin (30° - 45°) = 

2. tan (30°-^)= 4. tan (45° + 60°) = 6. cos (180° - 45°) = 
7. What do the following formulas become if a = /3 ? 

sin (a + p)= sin a cos /3 + cos a sin /3o . fa A- 3^— ^^^ ^ + ^^^ i^ . 

sin (06 — j8) = sin a cos |8 — cos a sin /3. 1 — tan a tan /3 

cos(a + /3)= coscccos j8 — sin a sin/?. , . ^x_ tanoc — tanj8 ^ 

cos (a — j3) = cos a cos /3 + sin a sin /3. 1 + tan a tan ^ 



VIII, § 68] TRIGONOMETRIC RELATIONS 97 

8. Complete the following formulas : 

sin 2 a cos a -f cos 2 a sin a = tan 2 a + tanoc _ 



9. 



sin 3 a cos a — cos 3 a sin a = 1 — tan 2 a tan a 

Prove sin 75° = V|^ ^ ^^^ ^^o ^ V3- 1 ^ ^^^ 750 ^ V3 + 1 



: sm a. 



2\/2 2V2 V3-1 

10. Given tan a = J, sin ^3 = j\, and a and j3 both positive acute angles, 
find the value of tan (a + ^) ; sin (a — /3) ; cos (a + /3) ; tan (a — /3). 

11. Prove that 

(a) cos (60° + a) + sin (30° + a) = cos a. 

(6) sin (60° + I?)- sin (60° -6) = sin 6>. 

(c) cos (30° + <?)- cos (30° - ^) = - sin 6. 

(d) cos (45° + d)+ cos (45° - 1?) = \/2 • cos ^. 

(e) sin^a+|Wsin^a-|^: 
(/) cos [« + -)+ cos (a--\ = \^S' cos a. 

(a) tan(45° + 0= ^ +tan^ ^ ,^. tan (45° - 6>)= ^ " ^^^^ .' 

^^ ^ ^ l-tan6> ^ ^ ^ ^ l+tani? 

12. By using the functions of 60° and 30° find the value of sin 90° ; 
cos 90°. 

13. Find in radical form the value of sin 15° ; cos 15° ; tan 15° ; 
sin 105° ; cos 105° ; tan 105°. 

14. If tan a = f , sin /3 = y\, and a is in the third quadrant while /3 is 
in the second, find sin (a ± ^); cos (« ± ^) ; tan (a ± /3). 

Prove the following identities : 

- - sin (a + /3) _ tan a + tan ^ ^ ^^ sin 2 a , cos 2 a _ gjv, s /v 

sin(a — /3) tana — tan j3 * sec a esc a 

^^ tancc-tan(ce-/3) ^ ^^^ ^ 19. (^) sin (i80° - $) = sin 0. 

1 4- tan a tan (a-^) (5) cos (180° -$) = - cos d. 

18. tan(^±45°) + ctn(6>T45°)=0. (c) tan (180° - ^) = - tan (9. 

20. cos (a + /3) cos (a — /3) = cos2 a — sin2 j3. 

21. sin (a -\- /3) sin (a — j3) = sin2 a - sin2 ^. 

22. ctn(a + ^) = "^^^^^^^-^. 23. ctn (« - « = ^^^^.^^^^^±1 . 

ctn a + ctn /8 ctn j8 — ctn a 

24. Prove Arc tan ^ + Arc tan i = 7r/4. 

[Hint : Let Arc tan ^ = x and Arc tan ^ = y. Then we wish to prove 
X + 1/ = 7r/4, which is true since tan (x + y)= l.~\ 

25. Prove Arc sin a + Arc cos a=:-ifO<a<l. 

26. Prove Arc sin y\ + Arc sin | = Arc sin ||. 

H 



98 PLANE TRIGONOMETRY [VIII, § 68 

27. Prove Arc tan 2 + Arc tan i = 7r/2. 

28. Prove Arc cos f + Arc cos (— j\) = Arc cos (— f|). 

29. Prove Arc tan j^- + Arc tan| = Arc tan J|. 

30. Find the value of sin [Arc sin | + Arc ctn|]. 

31. Find the value of sin [Arc sin a + Arc sin 6] if < a < 1, < 5 < 1. 

32. Expand sin {x -\- y -\- z) ; cos (x + y + a;). 
[Hint : X -{■ y -\- z =(x -{- y)-\- z.^ 

33. The area ^ of a triangle was computed from the formula 
A = \ah^ind. If an error e was made in measuring the angle ^, show that 
the corrected area A' is given by the relation ^' = J.(cos e + sin e ctn d). 

69. Functions of Double Angles. In this and the follow- 
ing articles (§§69-71) we shall derive from the addition 
formulas a variety of other relations which are serviceable in 
transforming trigonometric expressions. Since the formulas 
for sin (a + y8) and cos (a + ^) are true for all angles a and j8, 
they will be true when (3 = a. Putting /8 = cc, we obtain 

(1) sin 2 a = 2 sin a cos a, 

(2) cos 2 a = cos2 a — sin^ a. 
Since sin^ a + cos^ a = 1, we have also 

(3) cos 2 a = 1 - 2 sin2 a 

(4) = 2 cos2 a - 1. 

Similarly the formula for tan (a + fi) (which is true for all 
angles a, fi, and a+ ^ which have tangents) becomes, when /3= a, 

(5) tan2a= ^^^^^ , 
^ ^ l-tan2a 

which holds for every angle for which both members are defined. 
The above formulas should be learned in words. For ex- 
ample, formula (1) states that the sine of any angle equals 
twice the sine of half the angle times the cosine of half the 
angle. Thus sin 6 a: = 2 sin 3 a; cos 3 a;, 

2 tan 2 x 



tan 4 X ■■ 



cos X = cos2 - ~ sin2 -. 



tan2 2x' 
2 



VIII, § 70] TRIGONOMETRIC RELATIONS 99 

70. Functions of Half Angles. From (3), § 69, we have 

2 
Therefore 



2sin2 — = 1 — cos a. 



/^v . a , /I — cos a 
(^) sin- = ±^ -^ 

From (4), § 69, we have 

2 
Therefore 



2 cos2 - = 1 + cos a. 



/-x « , /I + COS a 
(0 cos-=±^/^^^^ 

Formulas (6) and (7) are at once seen to hold for all angles 
«. Now, if we divide formula (6) by formula (7), we obtain 



(8) tan|=±^/; 



1— COS a 



'1 H- cos a' 

which is true for all angles a except n • 180°, where n is any- 
odd integer. 

Example. Given sin ^ = — 3/5, cos J. negative ; find sin (^/2). 

Since the angle A is in the third quadrant, Ajl is in the second or 
fourth quadrant, and hence sin (^/2) may be either positive or negative. 
Therefore, since cos^ = — 4/5, we have 

2 V 2 VlO 10 

EXERCISES 

Complete the following formulas and state whether they are true for 
all angles : 

1. sin 2 a = 3. tan 2 a = 5. cos - = 

2. cos2a= (three forms). 4 sin - = 6. tan- = 

2 2 

7. In what quadrant is (9/2 if B is positive, less than 360°, and in the 
second quadrant ? third quadrant ? fourth quadrant ? 

8. Express cos 2 a in terms of cos 4 a. 

9. Express sin 6 x in terms of functions of 3 x. 



100 PLANE TRIGONOMETRY [VIII, § 70 

10. Express tan 4 a in terms of tan 2 a. 

11. Express tan 4 a in terms of cos 8 a. 

12. Express sinx in terms of f mictions of x/2. 

13. Explain why the formulas for sin x and cos x in terms of functions 
of 2 X have a double sign. 

14. From the functions of SO"" find those of 60°. 

15. From the functions of 60° find those of 30°. 

16. From the functions of 30° find those of 15°. 

17. From the functions of 15° find those of 7°. 5. 

18. Find the functions of 2 a if sin a = f and a is in the second 
quadrant. 

19. Find the functions of a/2 if cos a =—0.6 and a is in the third 
quadrant, positive, and less than 360°. 

20. Express sin 3 a in terms of sin a. [Hint : Sa = 2a -\- a.'] 

21. From the value of cos 45° find the functions of 22°. 5. 

22. Given sin a = — and a in the second quadrant. Find the values of 
(a) sin 2 a. (c) cos 2 a. (e) tan 2 a. 

(6) sin|. (d) cos^. (/) tan|. 

23. If tan 2 a = | find sin a, cos a, tan a if a is an angle in the third 
quadrant. 

Prove the following identities : 

24. L+^2^=ctn^. 27. 1 - cos 2 ^ + sin2 ^ ^^^^^^ 



25. 
26. 



sin a 2 1 + cos 2 + sin 2 <9 

rsin--cos-l =1 — sin^. 28. sin - -f- cos - = ± Vl + sin a. 

L.2 2J 22 



3s2^ + cos^-f 1 , „ ^_ ^ J. fir a\ 

. ^ . = ctn e, 29. seca + tana=tan( - + - ). 

sm 2 ^ + sm ^ V 4 2 / - 

30. 2 Arc cos X = Arc cos (2x2—1). 

31. 2 Arc cos x = Arc sin (2 xVl — x2). 



32. tan [2 Arc tan x] = ^^-. 34. tan [2 Arc sec x] = ^b ^ ^'^'^ " ^ - 

1 - X2 •- 2 - X2 

33. cos [2 Arc tan x] = ~" ^ • 35. cos (2 Arc sin a) = 1 — 2 a*. 

1 + x2 ^ ^ 

Solve the following equations ; 

36. cos 2 X + 5 sin x = 3. 40. sin2 2 x — sin2 x — \. 

37. cos2x — sinx = ^. 41. sin2x = 2cosx. 

38. sin 2 X cos X = sin X. 42. 2sin22x = 1 — cos2x. 

39. 2sin2x + sin22x = 2. 43. ctnx — csc2x — 1. 



VIII, § 71] TRIGONOMETRIC RELATIONS 101 

44. A flagpole 60 ft. high stands on a tower 49 ft. high. At what dis- 
tance from the foot of the tower will the flagpole and the tower subtend 
equal angles ? 

45. The dial of a town clock has a diameter of 10 ft. and its center is 
100 ft. above the ground. At what distance from the foot of the tower 
will the dial be most plainly visible ? [The angle subtended by the dial 
must be as large as possible.] 

71. Product Formulas. From § 68 we have 

sin (a -f- ^) = sin a cos ^ -f cos a sin ^, 
sin (a — /3) = sin a cos /3 — cos a sin y3. 

Adding, we get 

(1) sin (a + yS) + sin (« — /?) = 2 sin a cos (3. 

Subtracting, we have 

(2) sin (a + /?) — sin (a — /3) =2 cosa sin ^. 

Now, if we let a -f- ^ = P and a — /3 = Q, 

then « = ^±^, 1^ = ^^' 

2 ' ^ 2 

Therefore formulas (1) and (2) become 

P -4- O p — O 

sin P + sin Q = 2 sin ^ ^ cos ^, 

sin P — sin O = 2 cos ^-±-2 sin ^ ^ . 
^22 

Similarly, starting with cos (« + /S) and cos {a — f3) and per- 
forming the same operations, the following formulas result : 

P -\- O P — O 

cos P -\- COS = 2 cos — cos —, 

2 2 

P 4- O P — O 
cos P — COS Q = — 2 sin — -^— ^ sin — --^. 

2 2 

In words : 
the sum of two sines = 

twice sin (half sum) times cos (half difference), 
the difference of two sines = 

twice cos (half sum) times sin (half difference),* 
♦ The difference is taken,, first angle minus the second. 



102 PLANE TRIGONOMETRY [VIII, § 71 

the sum of two cosines = 

twice cos (half sum) times cos (half difference), 
the difference of two cosines = 

minus twice sin (half sum) times sin (half difference).^ 
Example 1. Prove that 

sin 3 X + sin x 

for all angles for which both members are defined. 

cos 3 X + cos a; _ 2 cos ^(3 x ■\- x) cos |(3 x — x) _ cos 2 x _ , ^ 
sin 3 X 4- sin X 2 sin i(3x + x) cos ^(3x — x) ^sin 2x 

Example 2. Reduce sin 4 x + cos 2 x to the form of a product. 
We may write this as sin 4x + sin (90^ — 2x), which is equal to 

2 sin 4x + 90^-2x ^^^ 4x-90° + 2x ^ ^ sin (45° + x) cos (3 x - 45°). 

EXERCISES 

Reduce to a product : 

1. sin 4 ^ — sin 2 d. 4. cos 2^4- sin 2 6. 7. cos 3 x + sin 5 x. 

2. cos <9 + cos 3 (9. 5. cos 3 (9 — cos 6 (9. 8. sin 20° — sin 60°. 

3. cos 6^+ cos 2^. 6. sin (x + Ax) — sin x. 
Show that 

9. sin 20° + sin 40° = cos 10°. ^^ sin 15° + sin 75° _ _ ^^^^ ^^^ 

10. cos 50° + cos 70° = cos 10^ ' sin 15° - sin 75° 

11. sin 75°-^ sin 15° ^^^^3Qo^ ^3^ MlL^i^^Ai = _ ctn 4 ^. 
cos 75° 4- cos 15° cos 3 ^ — cos 5 6 

Prove the f oUowino^ identities : 

14 



16. 
17. 



sin 4 « + sin 3 g _ , a ^^ sin a + sin j3 _ tan i (a + ^3) 

cos 3 a — cos 4 a 2 ' since — sin /3 tan i (a — /3) 

cos a + 2 cos 3 a + cos 5 a _ cos 3 a 



cos 3 a + 2 cos 5 cc + cos 7 a cos 5 a 



cosfx— cosj3 _ ___ tani(a + ^) - g sin (?i — 2) ^ + sin nO _ . ^ 

cosa + cos/3 ctni(oc — j8) ' cos (7i — 2) ^ — cos n^ 

Solve the following equations : 

19. cos ^ + cos 5 ^ = cos 3 6. 22. sin 4 ^ — sin 2 ^ = cos 3 0. 

20. sin (9 4- sin 5 <9 = sin 3 d. 23. cos 7 ^ — cos (9 = — sin 4 (9. 

21. sin 3 + sin 7 /9 = sin 5 (9. 

*The difference is taken, first angle minus the second. 



VIIL § 71] TRIGONOMETRIC RELATIONS 103 

MISCELLANEOUS EXERCISES 

1. Reduce to radians 05^ - 185^, — oOO\ 20^. 

2. Reduce to degrees tt, 3 tt, — 2 tt, 4 tt radians. 

3. Find sin (a — ^) and cos (a + ^) when it is given that a and ^ are 
positive and acute and tan « = | and sec ^ = y-. 

4. Find tan (a + j9) and tan (a — /3) when it is given that tan a = ^ 



6. Prove that sin 4 a = 4 sin a cos a — S sin^ a cos a, 
2 

Vo 



2 

6. Given sin^=^— , and in the second quadrant. Find sin2 6> 



cos 2 d, tan 2 ^. 

Prove the following identities ; 
7. sin2a=^i^^^. 9. sec2« = - "^'^'^ 



1 + tan2 a csc2 a — 2 

o o 1 — tan2 a </v ^ sin 2 a 

8. cos2a= 10. tana^ 



1 -h tan2 a 1 + cos 2 a 

11. sin (a + /3) cos ^ — cos (a + /3) sin /3 = sin «. 

12. sin 2 a + sin 2 /3 + sin 2 7 = 4 sin a sin /3 sin 7, if a + /3 + 7 = 180^^. 

1 + tan - 
cosa _ 2 

■ l-s'i'«~i_tan«' 



TABLES 

TO 

FOUR DECIMAL PLACES 



106 



Squares of Numbers 



[]Moving the decimal point on-e place in N requires a corresponding move of two 

places in N^] 



N 


F 


1 


2 


3 


4 


5 


6 


7 


8 


9 


0.0 


.0000 


.0001 


.0004 


.0009 ' .0016 


.0025 


.0036 


.0049 


.0064 


.0081 


0.1 
0.2 
0.3 

0.4 
0.5 
0.6 

0.7 
0.8 
0.9 


.0100 
.0400 
.0900 

.1600 
.2500 
.3600 

.4900 
.6400 
.8100 


.0121 
.0441 
.0961 

.1681 
.2601 
.3721 

.5041 
.6561 

.8281 


.0144 
.0484 
.1024 

.1764 
.2704 
.3844 

.5184 
.6724 
.8464 


.0169 
.0529 
.1089 

.1849 
.2809 
.3969 

.5329 

.6889 
.8649 


.0196 
.0576 
.1156 

.1936 
.2916 
.4096 

.5476 
.7056 
.8836 


.0225 
.0625 
.1225 

.2025 
.3025 
.4225 

.5625 
.7225 
.9025 


.0256 
.0676 
.1296 

.2116 
.3136 
.4356 

.5776 
.7396 
.9216 


.0289 
.0729 
.1369 

.2209 
.3249 
.4489 

.5929 
.7569 
.9409 


.0324 
.0784 
.1444 

.2304 
.3364 
.4624 

.6084 
.7744 
.9604 


.0361 
.0841 
.1521 

.2401 
.3481 
.4761 

.6241 
.7921 
.9801 


1.0 


1.000 


1.020 


1.040 


1.061 


1.082 


1.103 


1.124 


1.145 


1.166 


1.188 


1.1 
1.2 
1.3 

1.4 
1.5 
1.6 

1.7 

1.8 
1.9 


1.210 
1.440 
1.690 

1.960 
2.250 
2.560 

2.890 
3.240 
3.610 


1.232 
1.464 
1.716 

1.988 
2.280 
2.592 

2.924 
3.276 
3.648 


1.254 

1.488 
1.742 

2.016 
2.310 
2.624 

2.958 
3.312 
3.686 


1.277 
1.513 
1.769 

2.045 
2.341 
2.657 

2.993 
3.349 
3.725 


1.300 
1.538 
1.796 

2.074 
2.372 
2.690 

3.028 
3.386 
3.764 


1.323 
1.563 
1.823 

2.103 
2.403 
2.723 

3.063 
3.423 
3.803 


1.346 

1.588 
1.850 

2.132 
2.434 
2.756 

3.098 
3.460 
3.842 


1.369 
1.613 

1.877 

2.161 
2.465 

2.789 

3.133 
3.497 

3.881 


1.392 
1.638 
1.904 

2.190 
2.496 

2.822 

3.168 
3.534 
3.920 


1.416 
1.664 
1.932 

2.220 
2.528 
2.856 

3.204 
3.572 
3.960 


2.0 


4.000 


4.040 


4.080 


4.121 


4.162 


4.203 


4.244 


4.285 


4.326 


4.368 


2.1 
2.2 
2.3 

2.4 
2.5 
2.6 

2.7 

2.8 
2.9 


4.410 
4.840 
5.290 

5.760 
6.250 
6.760 

7.290 
7.840 
8.410 


4.452 
4.884 
5.336 

5.808 
6.300 
6.812 

7.344 
7.896 
8.468 


4.494 
4.928 
5.382 

5.856 
6.350 
6.864 

7.398 
7.952 
8.526 


4.537 
4.973 
5.429 

5.905 
6.401 
6.917 

7.453 
8.009 

8.585 


4.580 
5.018 
5.476 

5.954 
6.452 
6.970 

7.508 
8.066 
8.644 


4.623 
5.063 
5.523 

6.003 
6.503 
7.023 

7.563 
8.123 

8.703 


4.666 
5.108 
5.570 

6.052 
6.554 
7.076 

7.618 
8.180 
8.762 


4.709 
5.153 
5.617 

6.101 
6.605 
7.129 

7.573 
8.237 

8.821 


4.652 
5.198 
5.664 

6.150 
6.656 
7.182 

7.728 
8.294 

8.880 


4.796 
5.244 
5.712 

6.200 
6.708 
7.236 

7.784 
8.352 
8.940 


3.0 


9.000 


9.060 


9.120 


9.181 


9.242 


9.303 


9.364 


9.425 


9.486 


9.548 


3.1 
3.2 
3.3 

3.4 
3.5 
3.6 

3.7 
3.8 
3.9 


9.610 
10.24 
10.89 

11.56 
12.25 
12.96 

13.69 
14.44 
15.21 


9.672 
10.30 
10.96 

11.63 
12.32 
13.03 

13.76 
14.52 
15.29 


9.734 
10.39 
11.02 

11.70 
12.39 
13.10 

13.84 
14.59 
15.37 


9.797 
10.43 
11.09 

11.76 
12.46 
13.18 

13.91 
14.70 
15.44 


9.860 
10.50 
11.16 

11.83 
12.53 
13.25 

13.99 
14.75 
15.52 


9.923 
10.56 
11.22 

11.90 
12.60 
13.32 

14.06 
14.82 
15.60 


9.986 
10.63 
11.29 

11.97 
12.67 
13.40 

14.14 
14.90 
15.68 


10.05 
10.69 
11.36 

12.04 
12.74 
13.47 

14.21 
14.98 
15.76 


10.11 
10.76 
11.42 

12.11 
12.82 
13.54 

14.29 
15.05 
15.84 


10.18 
10.82 
11.49 

12.18 
12.89 
13.62 

14.26 
15.13 
15.92 


4.0 


16.00 


16.08 


16.16 


16.24 


16.32 


16.40 


16.48 


16.56 


16.65 


16.73 


4.1 
4.2 
4.3 

4.4 
4.5 
4.6 

4.7 
4.8 
4.9 


16.81 
17.64 
18.49 

19.36 
20.25 
21.16 

22.09 
23.04 
24.01 


16.89 
17.72 

18.58 

19.45 
20.34 
21.25 

22.18 
23.14 
24.11 


16.97 
17.81 
18.66 

19.54 
20.43 
21.34 

22.28 
23.23 
24.21 


17.06 
17.89 
1§.65 

19.62 
20.52 
21.44 

22.37 
23.33 
24.30 


17.14 
17.98 

18.84 

19.71 
20.61 
21.53 

22.47 
23.43 
24.40 


17.22 
18.06 
18.92 

19.80 
20.70 
21.62 

22.56 
23.52 
24.50 


17.31 
18.15 
19.01 

19.89 
20.79 
21.72 

22.66 
23.62 
24.60 


17.39 
18.23 
19.10 

19.98 

20.88 
21.81 

22.75 
23.72 
24.70 


17.47 
18.32 
19.18 

20.07 
20.98 
21.90 

22.85 
23.81 
24.80 


17.56 
18.40 
19.27 

20.16 
21.07 
22.00 

22.94 
23.91 
24.90 


5.0 


25.00 


25.10 


25.20 


25.30 


25.40 


25.50 


25.60 


25.70 


25.81 


25.91 



Squares of Numbers 



107 



[Moving the decimal point one place in N requires a corresponding move of two 

places in N^l 



N 


W 


1 


2 


3 


4 


5 


6 


7 


8 


9 


5.0 


25.00 


25.10 


25.20 


25.30 


25.40 


25.50 


25.60 


25.70 


25.81 


25.91 


5.1 
5.2 
5.3 

5.4 
5.5 
5.6 

5.7 

5.8 
5.9 

6.0 


26.01 
27.04 
28.09 

29.16 
30.25 
31.36 

32.49 
33.64 
34.81 


26.11 
27.14 
28.20 

29.27 
30.36 
31.47 

32.60 
33.76 
34.93 


26.21 
27.25 
28.30 

29.38 
30.47 
31.58 

32.72 
33.87 
35.05 


26.32 
27.35 
28.41 

29.48 
30.58 
31.70 

32.83 
33.99 
35.16 


26.42 
27.46 
28.52 

29.59 
30.69 
31.81 

32.95 
34.11 
35.28 


26.52 
27.56 
28.62 

29.70 
30.80 
31.92 

33.06 
34.22 
35.40 


26.63 
27.67 
28.73 

29.81 
30.91 
32.04 

33.18 
34.34 
35.52 


26.73 

27.77 
28.84 

29.92 
31.02 
32.15 

33.29 
34.46 
35.64 


26.83 

27.88 
28.94 

30.03 
31.14 
32.26 

33.41 
34.57 
35.76 


26.94 
27.98 
29.05 

30.14 
31.25 
32.38 

33.52 
34.69 

35.88 


36.00 


36.12 


36.24 


36.36 


36.48 


36.60 


36.72 


36.84 


36.97 


37.09 


6.1 
6.2 
6.3 

6.4 
6.5 
6.6 

6.7 
6.8 
6.9 

7.0 


37.21 
38.44 
39.69 

40.96 
42.25 
43.56 

44.89 
46.24 
47.61 


37.33 
38.56 
39.82 

41.09 
42.38 
43.69 

45.02 
46.38 
47.75 


37.45 
38.69 
39.94 

41.22 
42.51 
43.82 

45.16 
46.51 
47.89 


37.58 
38.81 
40.07 

41.34 
42.64 
43.96 

45.29 
46.65 
48.02 


37.70 
38.94 
40.20 

41.47 

42.77 
44.09 

45.42 
46.79 
48.16 


37.82 
39.06 
40.32 

41.60 
42.90 
44.22 

45.56 
46.92 
48.30 


37.95 
39.19 
40.45 

41.73 
43.03 
44.36 

45.70 
47.06 
48.44 


38.07 
39.31 
40.58 

41.86 
43.16 
44.49 

45.83 
47.20 

48.58 


38.19 
39.44 
40.70 

41.99 
43.30 
44.62 

45.97 
47.33 

48.72 


38.32 
39.56 
40.83 

42.12 
43.43 
44.76 

46.10 
47.47 

48.72 


49.00 


49.14 


49.28 


49.42 


49.56 


49.70 


49.84 


49.98 


50.13 


50.27 


7.1 
7.2 
7.3 

7.4 
7.5 
7.6 

7.7 
7.8 
7.9 


50.41 
51.84 
53.29 

54.76 
56.25 
57.76 

59.29 
60.84 
62.41 


50.55 
51.98 
53.44 

54.91 
56.40 
57.91 

59.44 
61.00 
62.57 


50.69 
52.13 
53.58 

55.06 
56.55 
58.06 

59.60 
61.15 
62.73 


50.84 
52.27 
53.73 

55.20 
56.70 

58.22 

59.75 
61.31 

62.88 


50.98 
'52.42 

53.88 

55.35 
56.85 
58.37 

59.91 
61.47 
63.04 


51.12 
52.56 
54.02 

55.50 
57.00 
58.52 

60.06 
61.62 
63.20 


51.27 
52.71 
54.17 

55.65 
57.15 

58.68 

60.22 
61.78 
63.36 


51.41 
52.85 
54.32 

55.80 
57.30 

58.83 

60.37 
61.94 
63.52 


51.55 
53.00 
54.46 

55.95 
57.46 

58.98 

60.53 
62.09 
63.68 


51.70 
53.14 
54.61 

56.10 
57.61 
59.14 

60.68 
62.25 
63.84 


8.0 64.00 


64.16 


64.32 


64.48 i 64.64 


64.80 


64.96 


65.12 


65.29 


65.45 


8.1 
8.2 
8.3 

8.4 
8.5 
8.6 

8.7 
8.8 
8.9 


65.61 
67.24 
68.89 

70.56 
72.25 
73.96 

75.69 
77.44 
79.21 


65.77 
67.40 
69.06 

70.73 
72.42 
74.13 

75.86 
77.62 
79.39 


65.93 
67.57 
69.22 

70.90 
72.59 
74.30 

76.04 
77.79 
79.57 


66.10 
67.73 
69.39 

71.06 
72.76 

74.48 

76.21 
77.97 
79.74 


66.26 
67.90 
69.56 

71.23 
72.93 
74.65 

76.39 
78.15 
79.92 


66.42 
68.06 
69.72 

71.40 
73.10 
74.82 

76.56 
78.32 
80.10 


66.59 
68.23 
69.89 

71.57 
73.27 
75.00 

76.74 
78.50 

80.28 


66.75 
68.39 
70.06 

71.74 
73.44 
75.17 

76.91 

78.68 
80.46 


66.91 
68.56 
70.22 

71.91 
73.62 
75.34 

77.08 
78.85 
80.64 


67.08 
68.72 
70.39 

72.08 
73.79 
75.52 

77.26 
79.03 
80.82 


9.0 


81.00 1 81.18 


81.36 


81.54 


81.72 


81.90 


82.08 


82.26 


82.45 


82.63 


9.1 
9.2 
9.3 

9.4 
9.5 
9.6 

9.7 
9.8 
9.9 


82.81 1 82.99 
84.64 : 84.82 
86.49 86.68 

88.36 88.55 
90.25 90.44 
92.16 92.35 

94.09 94.28 
96.04 96.24 
98.01 ! 98.21 


83.17 
85.00 
86.86 

88.74 
90.63 
92.54 

94.48 
96.43 
98.41 


83.36 
85.19 
87.05 

88.92 
90.82 
92.74 

94.67 
96.63 
98.60 


83.54 
85.38 
87.24 

89.11 
91.01 
92.93 

94.87 
96.83 
98.80 


83.72 
85.56 

87.42 

89.30 
91.20 
93.12 

95.06 
97.02 
99.00 


83.91 
85.75 
87.61 

89.49 
91.39 
93.32 

95.26 
97.22 
99.20 


84.09 
85.93 
87.80 

89.68 
91.58 
93.51 

95.45 
97.42 
99.40 


84.27 
86.12 
87.99 

89.87 
91.78 
93.70 

95.65 
97.61 
99.60 


84.46 
86.30 
88.17 

90.06 
91.97 
93.90 

95.84 
97.81 
99.80 



108 



Powers and Roots 



Squares and Cubes Square Roots and Cube Roots 



No. 


Square 


Cube 


Square 
EooT 


Cube 
EooT 


No. 


Square 


Cube 


Square 
KooT 


Cube 
KooT 


1 


1 


1 


1.000 


1.000 


61 


2,601 


132,651 


7.141 


3.708 


2 


4 


8 


1.414 


1.260 


52 


2,704 


140,608 


7.211 


3.733 


3 


9 


27 


1.732 


1.442 


53 


2,809 


148,877 


7.280 


3.756 


4 


16 


64 


2.000 


1.587 


54 


2,916 


157,464 


7.348 


3.780 


5 


25 


125 


2.236 


1.710 


55 


3,025 


166,375 


7.416 


3.803 


6 


36 


216 


2.449 


1.817 


56 


3,136 


175,616 


7.483 


3.826 


7 


49 


343 


2.646 


1.913 


57 


3,249 


185,193 


7.550 


3.849 


8 


64 


512 


2.828 


2.000 


58 


3,364 


195,112 


7.616 


3.871 


9 


81 


729 


3.000 


2.080 


59 


3,481 


205,379 


7.681 


3.893 


10 


100 


1,000 


3.162 


2.154 


60 


3,600 


216,000 


7.746 


3.915 


11 


121 


1,331 


3.317 


2.224 


61 


3,721 


226,981 


7.810 


3.936 


12 


144 


1,728 


3.464 


2.289 


62 


3,844 


238,328 


7.874 


3.958 


13 


169 


2,197 


3.606 


2.351 


63 


3,969 


250,047 


7.937 


3.979 


14 


196 


2,744 


3.742 


2.410 


64 


4,096 


262,144 


8.000 


4.000 


15 


225 


3,375 


3.873 


2.466 


65 


4,225 


274,625 


8.062 


4.021 


16 


256 


4,096 


4.000 


2.520 


66 


4,356 


287,496 


8.124 


4.041 


17 


289 


4,913 


4.123 


2.571 


67 


4,489 


300,763 


8.185 


4.062 


18 


324 


5,832 


4.243 


2.621 


68 


4,624 


314,432 


8.246 


4.082 


19 


361 


6,859 


4.359 


2.668 


69 


4,761 


328,509 


8.307 


4.102 


20 


400 


8,000 


4.472 


2.714 


70 


4,900 


343,000 


8.367 


4.121 


21 


441 


9,261 


4.583 


2.759 


71 


5,041 


357,911 


8.426 


4.141 


22 


484 


10,648 


4.690 


2.802 


72 


5,184 


373,248 


8.485 


4.160 


23 


529 


12,167 


4.796 


2.844 


73 


5,329 


389,017 


8.544 


4.179 


24 


576 


13,824 


4.899 


2.884 


74 


5,476 


405,224 


8.602 


4.198 


25 


625 


15,625 


5.000 


2.924 


75 


5,625 


421,875 


8.660 


4.217 


26 


676 


17,576 


5.099 


2.962 


76 


5,776 


438,976 


8.718 


4.236 


27 


729 


19,683 


5.196 


3.000 


77 


5,929 


456,533 


8.775 


4.254 


28 


784 


21,952 


5.292 


3.037 


78 


6,084 


474,552 


8.832 


4.273 


29 


841 


24,389 


5.385 


3.072 


79 


6,241 


493,039 


8.888 


4.291 


30 


900 


27,000 


5.477 


3.107 


80 


6,400 


512,000 


8.944 


4.309 


31 


961 


29,791 


5.568 


3.141 


81 


6,561 


531,441 


9.000 


4.327 


32 


1,024 


32,768 


5.657 


3.175 


82 


6,724 


551,368 


9.055 


4.344 


33 


1,089 


35,937 


5.745 


3.208 


83 


6,889 


571,787 


9.110 


4.362 


34 


1,156 


39,304 


5.831 


3.240 


84 


7,056 


592,704 


9.165 


4.380 


35 


1,225 


42,875 


5.916 


3.271 


85 


7,225 


614,125 


9.220 


4.397 


36 


1,296 


46,656 


6.000 


3.302 


86 


7,396 


636,056 


9.274 


4.414 


37 


1,369 


50,653 


6.083 


3.332 


87 


7,569 


658,503 


9.327 


4.431 


38 


1,444 


54,872 


6.164 


3.362 


88 


7,744 


681,472 


9.381 


4.448 


39 


1,521 


59,319 


6.245 


3.391 


89 


7,921 


704,969 


9.434 


4.465 


40 


1,600 


64,000 


6.325 


3.420 


90 


8,100 


729,000 


9.487 


4.481 


41 


1,681 


68,921 


6.403 


3.448 


91 


8,281 


753,571 


9.539 


4.498 


42 


1,764 


74,088 


6.481 


3.476 


92 


8,464 


778,688 


9.592 


4.514 


43 


1,849 


79,507 


6.557 


3.503 


93 


8,649 


804,357 


9.644 


4.531 


44 


1,936 


85,184 


6.633 


3.530 


94 


8,836 


830,584 


9.695 


4.547 


45 


2,025 


91,125 


6.708 


3.557 


95 


9,025 


857,375 


9.747 


4.563 


46 


2,116 


97,336 


6.782 


3.583 


96 


9,216 


884,736 


9.798 


4.579 


47 


2,209 


103,823 


6.856 


3.609 


97 


9,409 


912,673 


9.849 


4.595 


48 


2,304 


110,592 


6.928 


3.634 


98 


9,604 


941,192 


9.899 


4.610 


49 


2,401 


117,649 


7.000 


3.659 


99 


9,801 


970,299 


9.950 


4.626 


50 


2,500 


125,000 


7.071 


3.684 


100 


10,000 


1,000,000 


10.000 


4.642 



For a more complete table, see The jVIacmillan Tables, pp. 94-111. 



Important Constants 



109 



Certain Convenient Values for ?i = 1 to n = 10 



n 


\/n 


v,r 


'^n 


n I 


l//i ! 


Logic n 


1 


1.000000 


1.00000 


1.00000 


1 


1.0000000 


0.000000000 


2 


0.500000 


1.41421 


1.25992 


2 


0.5000000 


0.301029996 


3 


0.333333 


1.73205 


1.44225 


6 


0.1666667 


0.477121255 


4 


0.250000 


2.00000 


1.58740 


24 


0.0416667 


0.602059991 


5 


0.200000 


2.23607 


1.70998 


120 


0.0083333 


0.698970004 


6 


0.166667 


2.44949 


1.81712 


720 


0.0013889 


0.778151250 


7 


0.142857 


2.64575 


1.91293 


5040 


0.0001984 


0.845098040 


8 


0.125000 


2.82843 


2.00000 


40320 


0.0000248 


0.903089987 


9 


0.111111 


3.00000 


2.08008 


362880 


0.0000028 


0.954242509 


10 


0.100000 


3.16228 


2.15443 


3628800 


0.0000003 


1.000000000 



Logarithms of Important Constants 



n = NUMBER 


Yaltie of n 


LoGio n 


IT 


3.14159265 


0.49714987 


l-f-TT 


0.31830989 


9.50285013 


7r2 


9.86960440 


0.99429975 


VtF 


1.77245385 


0.24857494 


c = Napierian Base 


2.71828183 


0.43429448 


M= logio e 


0.43429448 


9.63778431 


l-HiV/=logelO 


2.30258509 


0.36221569 


180 -r- TT = degrees in 1 radian 


57.2957795 


1.75812262 


TT -T- 180 = radians in 1° 


0.01745329 


8.24187738 


TT -T- 10800 = radians in V 


0.0002908882 


6.46372613 


IT -^ 648000 = radians in 1" 


0.000004848136811095 


4.68557487 


sin 1" 


0.000004848136811076 


4.68557487 


tan 1" 


0.000004848136811152 


4.68557487 


centimeters in 1 ft. 


30.480 


1.4840158 


feet in 1 cm. 


0.032808 


8.5159842 


inches in 1 m. 


39.37 (exact legal value) 


1.5951654 


pounds in 1 kg. 


2.20462 


0.3433340 


kilograms in 1 lb. 


0.453593 


9.65666()0 


g (average value) 


32.16 ft. /sec/sec. 


1.5073 




= 981 cm. /sec. /sec. 


2.9916690 


weight of 1 cu. ft. of water 


62.425 lb. (max. density) 


1.7953586 


weight of 1 cu. ft. of air 


0.0807 lb. (at 32° F.) 


8.907 


cu. in. in 1 (U. S.) gallon 


231 (exact legal value) 


2.3636120 


ft. lb. per sec. in 1 H. P. 


550. (exact legal value) 


2.7403627 


kg. m. per sec. in 1 H. P. 


76.0404 


1.8810445 


watts in 1 H. P. 


745.957 


2.8727135 



no 










Four Place Logarithms 






N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


12 3 


4 5 6 


7 8 9 


10 


0000 


0043 


0086 


0128 


0170 


0212 


0253 


0294 


0334 


0374 


4 812 


17 21 25 


29 33 37 


11 
12 
13 

14 
15 

16 

17 

18 
19 


0414 
0792 
1139 

1461 
1761 
2041 

2304 

2553 

2788 


0453 

0828 
1173 

1492 

1790 
2068 

2330 
2577 
2810 


0492 
0864 
1206 

1523 

1818 
2095 

2355 

2601 
2833 


0531 
0899 
1239 

1553 

1847 
2122 

2380 
2625 
2856 


0569 
0934 
1271 

1584 

1875 
2148 

2405 
2648 

2878 


0607 
0969 
1303 

1614 
1903 
2175 

2430 
2672 
2900 


0645 
1004 
1335 

1644 
1931 
2201 

2455 
2695 
2923 


0682 
1038 
1367 

1673 
1959 

2227 

2480 
2718 
2945 


0719 
1072 
1399 

1703 

1987 
2253 

2504 
2742 
2967 


0755 
1106 
1430 

1732 
2014 
2279 

2529 
2765 
2989 


4 8 11 
3 7 10 
3 6 10 

3 6 9 
3 6 8 
3 5 8 

2 5 7 
2 5 7 

2 4 7 


15 If, 23 
14 17 21 
13 16 19 

12 15 18 
11 14 17 
11 13 16 

10 12 15 
9 12 14 
9 1113 


26 30 34 
24 28 31 
23 26 29 

21 24 27 
20 22 25 
18 21 24 

17 20 22 
16 19 21 
16 18 20 


20 


3010 


3032 


3054 


3075 


3096 


3118 


3139 


3160 


3181 


3201 


2 4 6 


8 11 13 


15 17 19 


21 
22 
23 

24 
25 

26 

27 
28 
29 


3222 
3424 
3617 

3802 
3979 
4150 

4314 
4472 
4624 


3243 
3444 
3636 

3820 
3997 
4166 

4330 

4487 
4639 

4786 


3263 
3464 
3655 

3838 
4014 
4183 

4346 
4502 
4654 

4800 


3284 
3483 
3674 

3856 
4031 
4200 

4362 
4518 
4669 


3304 
3502 
3692 

3874 
4048 
4216 

4378 
4533 
4683 


3324 
3522 
3711 

3892 
4065 
4232 

4393 
4548 
4698 


3345 
3541 
3729 

3909 
4082 
4249 

4409 
4564 
4713 


3365 
3560 
3747 

3927 
4099 
4265 

4425 
4579 
4728 


3385 
3579 
3766 

3945 
4116 
4281 

4440 
4594 
4742 


3404 
3598 
3784 

3962 
4133 

4298 

4456 
4609 

4757 


2 4 6 
2 4 6 
2 4 6 

2 4 5 
2 4 5 
2 3 5 

2 3 5 
2 3 5 
1 3 4 


8 10 12 
8 10 12 
7 9 11 

7 9 11 
7 9 10 
7 8 10 

6 8 9 
6 8 9 
6 7 9 


14 16 18 
14 16 17 
13 15 17 

12 14 16 
12 14 16 
11 13 15 

11 12 14 
11 12 14 
10 12 13 


30 


4771 


4814 


4829 


4843 


4857 


4871 


4886 


4900 


13 4 


6 7 9 


10 11 13 


31 
32 
33 

34 
35 

36 

37 
38 
39 


4914 
5051 
5185 

5315 
5441 
5563 

5682 
5798 
5911 


4928 
5065 
5198 

5328 
5453 
5575 

5694 

5809 
5922 


4942 
5079 
5211 

5340 
5465 

5587 

5705 
5821 
5933 


4955 
5092 
5224 

5353 

5478 
5599 

5717 

5832 
5944 


4969 
5105 
5237 

5366 
5490 
5611 

5729 
5843 
5955 


4983 
5119 
5250 

5378 
5502 
5623 

5740 

5855 
5966 


4997 
5132 
5263 

5391 
5514 
5635 

5752 

5866 
5977 


5011 
5145 
5276 

5403 
5527 
5647 

5763 

5877 
5988 


5024 
5159 
5289 

5416 
5539 
5658 

5775 

5888 
5999 


5038 
5172 
5302 

5428 
5551 
5670 

5786 
5899 
6010 


1 3 4 
13 4 
13 4 

12 4 
12 4 
12 4 

12 4 

1 2 3 
1 2 3 


5 7 8 
5 7 8 
5 7 8 

5 6 8 
5 6 7 
5 6 7 

5 6 7 
5 6 7 
4 5 7 


10 11 12 
91112 
9 1112 

910 11 
9 10 11 
8 1011 

8 911 
8 9 10 
8 9 10 


40 


6021 


6031 


6042 


6053 


6064 


6075 


6085 


6096 


6107 


6117 


12 3 

12 3 
12 3 
12 3 

12 3 

12 3 
12 3 

12 3 
12 3 
12 3 


4 5 6 


8 9 10 


41 
42 
43 

44 
45 

46 

47 
48 
49 


6128 
6232 
6335 

64a5 
6532 

6628 

6721 
6812 
6902 


6138 
6243 
6345 

6444 
6542 
6637 

6730 

6821 
6911 


6149 
6253 
6355 

6454 
6551 
6646 

6739 
6830 
6920 


6160 
6263 
6365 

6464 
6561 
6656 

6749 
6839 
6928 


6170 
6274 
6375 

6474 
6571 
6665 

6758 
6848 
6937 


6180 
6284 
6385 

6484 
6580 
6675 

6767 
6857 
6946 


6191 
6294 
6395 

6493 
6590 
6684 

6776 
6866 
6955 


6201 
6304 
6405 

6503 
6599 
6693 

6785 
6875 
6964 


6212 
6314 
6415 

6513 
6609 
6702 

6794 

6884 
6972 


6222 
6325 
6425 

6522 
6618 
6712 

6803 
6893 
6981 


4 5 6 
4 5 6 
4 5 6 

4 5 6 
4 5 6 
4 5 6 

4 5 6 
4 5 6 
4 4 5 


7 8 9 

7 8 9 
7 8 9 

7 8 9 
7 8 9 
7 7 8 

7 7 8 
7 7 8 
6 7 8 


50 


6990 


6998 


7007 


7016 


7024 


7033 


7042 


7050 


7059 


7067 


12 3 


3 4 5 


6 7 8 


51 
52 
53 

54 


7076 
7160 
7243 

7324 


7084 
7168 
7251 

7332 


7093 
7177 
7259 

7340 


7101 

7185 
7267 

7348 


7110 
7193 
7275 

7356 


7118 
7202 

7284 

7364 


7126 
7210 
7292 

7372 


7135 

7218 
7300 

7380 


7143 

7226 
7308 

7388 


7152 

7235 
7316 

739(3 


12 3 
12 3 
12 2 

12 2 


3 4 5 
3 4 5 
3 4 5 

3 4 5 


6 7 8 
6 7 7 
6 6 7 

6 6 7 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


12 2 


4 5 6 


7 8 9 



The proportional parts are stated in full for every tenth at the right-hand side. 
The logarithm of any number of four significant figures can be read directly by add- 













Four Place Logarithms 




111 


N 





1 


2 


3 


4 


5 


6 


7 


8 


9 


1 2 3 


4 5 6 


7 8 9 


55 

56 

57 

58 
59 


7404 
7482 

7559 
7634 
770^) 


7412 
7490 

7566 
7642 
7716 


7419 
7497 

7574 
7649 
7723 


7427 
7505 

7582 
7657 
7731 


743,") 
7513 

7589 
76(>4 

7738 


7443 
7520 

7597 
7672 
7745 


7451 

7528 

7604 
7679 

7752 


7459 
7536 

7612 

7686 
7760 


74()6 
7543 

7619 
7694 
7767 


7474 
7551 

7()27 
7701 
7774 


1 2 2 
12 2 

1 1 2 
112 
112 


3 4 5 
3. 4 5 

3 4 5 
3 4 4 
3 4 4 


5 6 7 

5 6 7 

6 6 7 
5 6 7 

5 6 7 


60 


7782 


7789 


7796 


7803 


7810 


7818 


7825 


7832 


7839 


7846 


1 1 2 


3 4 4 


5 6 6 


61 
62 
63 

64 
65 

67 
68 
69 


7853 
7924 
7993 

8062 
8129 
8195 

8261 
8325 

8388 


7860 
7931 
8000 

8069 
8136 
8202 

8267 
8331 
8395 


7868 
7938 
8007 

8075 
8142 
8209 

8274 
8338 
8401 


7875 
7M5 
8014 

8082 
8149 
8215 

8280 
8344 
8407 


7882 
7952 
8021 

8089 
8156 
8222 

8287 
8351 
8414 


7889 
7959 
8028 

8096 
8162 
8228 

8293 
8357 
8420 


7896 
1966 
8035 

8102 
8169 
8235 

8299 
8363 
8426 


7903 7910! 7917 
7973 7980,7987 
8041 8048| 8055 

8109 8116' 8122 
8176, 8182 8189 
8241 8248 8254 

8306* 8312 8319 
8370; 8376 8382 
8432 8439 8445 


112 
112 
112 

112 
1 1 2 
112 

112 
1 1 2 
1 1 2 


3 3 4 
3 3 4 
3 3 4 

3 3 4 
3 3 4 
3 3 4 

3 3 4 
3 3 4 
3 3 4 


5 6 6 
5 5 6 
5 5 6 

5 5 6 
5 5 6 
5 5 6 

5 5 6 
4 5 6 
4 5 6 


70 


8451 


^57 1 8463' 8470| 


8476 


8482 


8488 


849418500 8506 


1 1 2 


3 3 4 


4 5 6 


71 
72 
73 

74 
75 

76 

77 
78 > 
79' 


8513 
8573 
8633 

8692 
8751 
8808 

8865 
8921 
8976 


8519 
8579 
8639 

8698 
8756 
8814 

8871 
8927 
8982 


8525 
8585 
8645 

8704 
8762 
8820 

8876 
8932 

8987 


8531 
8591 
8651 

8710 

8768 
8825 

8882 
8938 
8993 


8537 
8597 
8657 

8716 
8774 
8831 

8887 
8943 
8998 


8543 
8603 
8663 

8722 
8779 
8837 

8893 
8949 
9004 


8549 
8609 
8669 

8727 
8785 
8842 

8899 
8954 
9009 


8555' 8561 8567 
8615' 8621 8627 
8675, 8681 8686 

8733 8739 8745 
8791 8797,8802 
8848 8854, 8859 

8904 8910' 8915 
89(30 8965 8971 
9015 9020 9025 


112 
1 1 2 
112 

112 
112 
112 

1 1 2 
1 1 2 
112 


3 3 4 
3 3 4 
2 3 4 

2 3 4 
2 3 3 
2 3 3 

2 3 3 
2 3 3 
2 3 3 


4 5 6 
4 5 6 
4 5 5 

4 5 5 
4 5 5 
4 4 5 

4 4 5 
4 4 5 
4 4 5 


80 


V)031 


9036 


9042! 9047 


9053 


9058 


9063 


9069 9074 9079 


112 


2 3 3 


4 4 5 


81 
82 
83 

84 
85 

86 

87 
88 
89 


908,5 
9138 
9191 

19243 
19294 
9345 

9395 
9445 
9494 


9090 
9143 
9196 

9248 
9299 
9350 

9400 
9450 
9499 


909(3 
9149 
9201 

9253 
9304 
9355 

9405 
9455 
9504 


9101 
9154 
9206 

9258 
9309 
9360 

^10 
9460 
9509 


9106 
9159 
9212 

9263 
9315 
9365 

9415 
9465 
9513 


9112 
9165 
9217 

9269 
9320 
9370 

9420 
9469 
9518 


9117 
9170 
9222 

9274 
9325 
9375 

9425 
9474 
9523 


9122 
9175 
9227 

9279 
9330 
9380 

9430 
9479 

9528 


9128 
9180 
9232 

9284 
9335 
9385 

9435 
9484 
9533 


9133 
9186 
9238 

9289 
9340 
9390 

9440 

9489 
9538 


112 
112 
112 

112 
112 
1 1 2 

112 
Oil 
1 1 


2 3 3 
2 3 3 
2 3 3 

2 3 3 
2 3 3 
2 3 3 

2 3 3 
2 2 3 
2 2 3 


4 4 5 
4 4 5 
4 4 5 

4 4 5 
4 4 5 
4 4 5 

4 4 5 
3 4 4 
3 4 4 


90 


9542 


9547 


9552 


9557 


9562 


9566 


9571 


9576 


9581 


9586 


Oil 


2 2 3 


3 4 4 


91 
92 
93 

94 
95 

96 

97 
98 
99 


9590 
9638 
9685 

9731 
i9777 
9823 

9868 
9912 
9956 


9595 
9f)43 
9689 

9736 

9782 
9827 

9872 
9917 
9961 


9600 
9647 
9694 

9741 

9786 
9832 

9877 
9921 
9965 


9605 
9652 
9699 

9745 
9791 
9836 

9881 

9926 

19969 


9609 
9657 
9703 

9750 
9795 
9841 

9886 
9930 
9974 


9614 
9661 

9708 

9754 
9800 
9845 

9890 
9934 
9978 


9619 
9666 
9713 

9759 
9805 
9850 

9894 
9939 
9983 


9624 
9671 
9717 

9763 
9809 
9854 

9899 
9943 

9^)87 


9628 
9675 
9722 

9768 
9814 
9859 

9903 
9948 
9991 


9633 
9680 
9727 

9773 
9818 
9863 

9908 
9952 
9996 


Oil 
1 1 
1 1 

1 1 
1 1 
1 1 

Oil 
1 1 
1 1 


2 2 3 
2 2 3 
2 2 3 

2 2 3 
2 2 3 
2 2 3 

2 2 3 
2 2 3 
2 2 3 


3 4 4 
3 4 4 
3 4 4 

3 4 4 
3 4 4 
3 4 4 

3 4 4 
3 3 4 
3 3 4 


N ' 


1 


2 


3 


4 


5 


6 


7 


8 


9 


12 3 


4 5 6 


7 8 9 



jng the proportional part corresponding to the fourth figure to the tabular number 
corresponding to the ^st three figures. There may he an error of 1 in the last place. 



112 



Four Place Trigonometric Functions 



[Characteristics of Logarithms omitted — 


determine by the usual rule from the value] 


Radians 


Deg-rees 


Sine 


Tangent 


Cotangent 


Cosine 






XV.AJ^X^Vx^ o 


-M^ JZiXjfJJmXU-MiiiJ 


Value 


Logio 


Value Logio 


Value 


Logio 


Value 


Log-io 






.0000 
.0029 


0°00' 

10 


.0000 
.0029 




0000 






1.0000 


.0000 


90° 00' 

50 


1.5708 
1.5679 


.4637 


.0029 .4637 


343.77 


.5363 


LOOOO 


'.0000 


.0058 


20 


.0058 


.7648 


.0058 .7648 


171.89 


.2352 


1.0000 


.0000 


40 


1.5650 


.0087 


30 


.0087 


.9408 


.0087 .9409 


114.59 


.0591 


i.ooa) 


.0000 


30 


1.5621 


.0116 


40 


.0116 


.0658 


.0116 .0658 


85.940 


.9342 


.9999 


.0000 


20 


1.5592 


.0145 


50 


.0145 


.1627 


.0145 .1627 


68.750 


.8373 


.9999 


.0000 


10 


1.5563 


.0175 


1^00' 


.0175 


.2419 


.0175 .2419 


57.290 


.7581 


.9998 


.9999 


89° 00' 


1.5533 


.0204 


10 


.0204 


.3088 


.0204 .3089 


49.104 


.6911 


.9998 


.9999 


50 


1.5504 


.0233 


20 


.0233 


.3668 


.0233 .3669 


42.964 


.6331 


.9997 


.9999 


40 


1.5475 


.0262 


30 


.0262 


.4179 


.0262 .4181 


38.188 


.5819 


.9997 


.9999 


30 


1.5446 


.0291 


40 


.0291 


.4637 


.0291 .4638 


34.368 


.5362 


.9996 


.9998 


20 


1.5417 


.0320 


50 


.0320 


.5050 


.0320 .5053 


31.242 


.4947 


.9995 


.9998 


10 


1.5388 


.0349 


2^00' 


.0349 


.5428 


.0349 .5431 


28.636 


.4569 


.9994 


.9997 


88° 00 


1.5359 


.0378 


10 


.0378 


.5776 


.0378 .5779 


26.432 


.4221 


.9993 


.9997 


50 


1.5330 


.0407 


20 


.0407 


.6097 


.0407 .6101 


24.542 


.3899 


.9992 


.9996 


40 


1.5301 


.0436 


30 


.0436 


.6397 


.0437 .6401 


22.904 


.3599 


.99^)0 


.9996 


30 


1.5272 


.0465 


40 


.0465 


.6677 


.0466 .6682 


21.470 


.3338 


.9989 


.9995 


20 


1.5243 


.0495 


50 


.0494 


.6940 


.0495 .6945 


20.206 


.3055 


.9988 


.9995 


10 


1.5213 


.0524 


3^00' 


.0523 


.7188 


.0524 .7194 


19.081 


.2806 


.9986 


.9994 


87° 00' 


1.5184 


.0553 


10 


.0552 


.7423 


.0553 .7429 


18.075 


.2571 


.9985 


.9993 


50 


1.5155 


.0582 


20 


.0581 


.7645 


.0582 .7652 


17.169 


.2348 


.9983 


.9993 


40 


1.5126 


.0611 


30 


.0610 


.7857 


.0612 .7865 


16.350 


.2135 


.9981 


.9992 


30 


1.5097 


.0640 


40 


.0640 


.8059 


.0641 .8067 


15.605 


.1933 


.9980 


.9991 


20 


1.5068 


.0669 


50 


.0669 


.8251 


.0670 .8261 


14.924 


.1739 


.9978 


.9990 


10 


1.5039 


.0698 


4° 00' 


.0698 


.843() 


.0699 .8446 


14.301 


.1554 


.9976 


.9989 


86° 00 


1.5010 


.0727 


10 


.0727 


.8613 


.0729. .8624 


13.727 


.1376 


.9974 


.9989 


50 


1.4981 


.0756 


20 


.0756 


.8783 


.0758 .8795 


13.197 


.1205 


.9971 


.9988 


40 


1.4952 


.0785 


30 


.0785 


.8946 


.0787 .8%0 


12.706 


.1040 


.9969 


.9987 


30 


1.4923 


.0814 


40 


.0814 


.9104 


.0816 .9118 


12.251 


.0882 


.9967 


.9986 


20 


1.4893 


.0844 


50 


.0843 


.9256 


.0846 .9272 


11.826 


.0728 


.9964 


.9985 


10 


1.4864 


.0873 


5^00' 


.0872 


.9403 


.0875 .9420 


11.430 


.0580 


.9962 


.9983 


85° 00' 


1.4835 


.0902 


10 


.0901 


.9545 


.0904 .9563 


11.059 


.0437 


.9959 


.9982 


50 


1.4806 


.0931 


20 


.0929 


.9682 


.0934 .9701 


10.712 


.0299 


.9957 


.9981 


40 


1.4777 


.0960 


30 


.0958 


.9816 


.0963 .9836 


10.385 


.0164 


.9954 


.9980 


30 


1.4748 


.0989 


40 


.0987 


.9945 


.0992 .9966 


10.078 


.0034 


.9951 


.9979 


20 


1.4719 


.1018 


50 


.1016 


.0070 


.1022 .0093 


9.7882 


.9907 


.9948 


.9977 


10 


1.4690 


.1047 


6° 00' 


.1045 


.0192 


.1051 .0216 


9.5144 


.9784 


.9945 


.9976 


84° 00' 


1.4661 


.1076 


10 


.1074 


.0311 


.1080 .0336 


9.2553 


.9664 


.9942 


.9975 


50 


1.4632 


.1105 


20 


.1103 


.0426 


.1110 .0453 


9.0098 


.9547 


.9939 


.9973 


40 


1.4603 


.1134 


30 


.1132 


.0539 


.1139 .0567 


8.7769 


.9433 


.9936 


.9972 


30 


1.4573 


.1164 


40 


.1161 


.0648 


.1169 .0678 


8.5555 


.9322 


.9932 


.9971 


20 


1.4544 


.1193 


50 


.1190 


.0755 


.1198 .0786 


8.3450 


.9214 


.9929 


.9969 


10 


1.4515 


.1222 


7° 00' 


.1219 


.0859 


.1228 .0891 


8.1443 


.9109 


.9925 


.9968 


83° 00' 


1.4486 


.1251 


10 


.1248 


.0961 


.1257 .0995 


7.9530 


.9005 


.9922 


.9966 


50 


1.4457 


.1280 


20 


.1276 


.1060 


.1287 .109(5 


7.7704 


.8904 


.9918 


.9964 


40 


1.4428 


.1309 


30 


.1305 


.1157 


.1317 .1194 


7.5958 


.8806 


.9914 


.9963 


30 


1.4399 


.1338 


40 


.1334 


.1252 


.1346 .1291 


7.4287 


.8709 


.9911 


.9961 


20 


1.4370 


.1367 


50 


.1363 


.1345 


.1376 .1385 


7.2687 


.8615 


.9907 


.9959 


10 


1.4341 


.1396 


8° 00' 


.1392 


.1436 


.1405 .1478 


7.1154 


.8522 


.9903 


.9958 


82° 00' 


1.4312 


.1425 


10 


.1421 


.1525 


.1435 .1569 


6.9682 


.8431 


.9899 


.9f)56 


50 


1.4283 


.1454 


20 


.1449 


.1612 


.1465 .1658 


6.8269 


.8342 


.9894 


.9954 


40 


1.4254 


.1484 


30 


.1478 


.1697 


.1495 .1745 


6.6912 


.8255 


.9890 


.9952 


30 


1.4224 


.1513 


40 


.1507 


.1781 


.1524 .1831 


6.5606 


.8169 


.9886 


.9950 


20 


1.4195 


.1542 


50 


.1536 


.1863 


.1554 .1915 


6.4348 


.8085 


.9881 


.9948 


10 


1.4166 


.1571 


9^00' 


.1564 


.1943 


.1584 .1997 


6.3138 


.8003 


.9877 


.9946 


81° 00' 


1.4137 






Value 


Lo?io 


Value Loff^o 


Value 


Logio 


Value 


Logio 


Degrees 


Radians 


1 




Cosine 


Cotangent 


Tangent 


Sine 







Four Place Trigonometric Functions 113 

[Cbaractoristics of Loirarithras omitted — determine by the usual rule from the value] 



Radians 


Degrees 


Si. 


SE 


Tangent 


COTAN 


GENT 


Cosine 










Value 


Logio 


Value 


Logio 


Value 


1-ogio 


Value Logio 






.1571 


9^00 


.1564 


.1943 


.15S4 


.1997 


6.3138 


.8003 


.9877 .9946 


81^00' 


1.4137 


.KMX) 


10 


.1593 


.2022 


.1614 


.2078 


6.1970 


.7922 


.9872 .9944 


50 


1.4108 


.1(529 


20 


.1622 


.2100 


.1(>44 


.2158 


6.0844 


.7842 


.9868 .9942 


40 


1.4079 


.IimS 


30 


.1650 


.217() 


.1673 


.223() 


5.9758 


.77(>4 


.9863 .9940 


30 


1.4050 


.U>87 


40 


.1(>79 


.2251 


.1703 


.2313 


5.8708 


.7687 


.9858 .9938 


20 


1.4021 


.1716 


50 


.1708 


.2324 


.1733 


.2389 


5.7694 


.7611 


.9853 .9936 


10 


1.3992 


.1745 


10^00 


.1736 


.2397 


.1763 


.2463 


5.6713 


.7537 


.9848 .9934 


80° 00' 


1.3963 


.1774 


10 


.1765 


.2468 


.1793 


.2536 


5.5764 


.7464 


.9^43 .9931 


50 


1.3934 


.1804 


20 


.171U 


.25:38 


.1823 


.2609 


5.4845 


.7391 


.9838 .9929 


40 


1.3904 


.1833 


30 


.1822 


.2606 


.1853 


.2680 


5.3955 


.7320 


.9833 .9927 


30 


1.3875 


.18!)2 


40 


.1851 


.2674 


.1883 


.2750 


5.3093 


.7250 


.9827 .9924 


20 


1.3846 


.1891 


50 


.1880 


.2740 


.1914 


.2819 


5.2257 


.7181 


.9822 .9922 


10 


1.3817 


.1920 


11^00 


.1908 


.2806 


.1944 


.2887 


5.1446 


.7113 


.9816 .9919 


79° 00' 


1.3788 


.15U9 


10 


.1937 


.2870 


.1974 


.2953 


5.0658 


.7047 


.9811 .9917 


50 


1.3759 


.1978 


20 


.1965 


.2934 


.2004 


.3020 


4.9894 


.6980 


.9805 .9914 


40 


1.3730 


.2007 


30 


.1994 


.2<)97 


.2035 


.3085 


4.9152 


.6915 


.9799 .9912 


30 


1.3701 


.2036 


40 


.2022 


.3058 


.2065 


.3149 4.8430 


.6851 


.9793 .9909 


20 


1.3672 


.2065 


50 


.2051 


.3119 


.2095 


.3212 j 4.7729 


.6788 


.9787 .9907 


10 


1.3643 


.2094 


12^00 


.2079 


.3179 


.2126 


.3275 4.7046 


.6725 


.9781 .9904 


78^00' 


1.3614 


.2123 


10 


.2108 


.32:38 


.2156 


.3336 4.6382 


.6664 


.9775 .9901 


50 


1.3584 


.2153 


20 


.21:36 


.3296 


.2186 


.3397 


4.5736 


.6603 


.9769 .9899 


40 


1.3555 


.2182 


30 


.21(>4 


.3353 


.2217 


.3458 


4.5107 


.6542 


.9763 .9896 


30 


1.3526 


.2211 


40 


.2193 


.3410 


.2247 


.3517 


4.4494 


.6483 


.9757 .9893 


20 


1.3497 


.2240 


50 


.2221 


.3466 


.2278 


.3576 


4.3897 


.6424 


.9750 .9890 


10 


1.3468 


.2269 


13^00 


.2250 


.3521 


.2309 


.3634 


4.3315 


.6366 


.9744 .9887 


77° 00' 


1.3439 


.2298 


10 


.2278 


.3575 


.2339 


.3691 


4.2747 


.6309 


.9737 .9884 


50 


1.3410 


.2327 


20 


.2306 


.3629 


.2370 


.3748 


4.2193 


.6252 


.9730 =9881 


40 


1.3381 


!235(j 


30 


.2334 


.3682 


.2401 


.3804 


4.1653 


.6196 


.9724 .9878 


30 


1.3352 


.2385 


40 


.2363 


.3734 


.2432 


.3859 i 4.1126 


.6141 


.9717 .9875 


20 


1.3323 


.2414 


50 


.2391 


.3786 


.2462 


.3914 


4.0611 


.6086 


.9710 .9872 


10 


1.3294 


.2443 


14^00' 


.2419 


.3837 


.2493 


.3968 


4.0108 


.6032 


.9703 .9869 


76° 00' 


1.3265 


.2473 


10 


.2447 


.3887 


.2524 


.4021 


3.9(317 


.5979 


.9696 .9866 


50 


1.3235 


.2502 


20 


.2476 


.3937 


.2555 


.4074 


3.9136 


.5926 1 .9689 .9863 


40 


1.3206 


.2531 


30 


.2504 


.3986 


.2586 


.4127 


3.8667 


.5873 1 .9681 .9859 


30 


1.3177 


.25<'30 


40 


.2.532 


.4035 .2617 


.4178 


3.8208 


.5822 .9674 .9856 


20 


1.3148 


.2589 


50 


.2560 


.4083 


.2(>48 


.4230 


3.7760 


.5770 .9667 .9853 


10 


1.3119 


.2618 


15^00 


.2588 


.4130 


.2679 


.4281 


3.7321 


.5719 .9659 .9849 


75° 00' 


1.3090 


.2647 


10 


.2616 


.4177 


.2711 


.4331 


3.6891 


.5669 .9652 .9846 


50 


1.3061 


.2676 


20 


.2644 


.4223 


.2742 


.4381 


3.6470 


.5619 .9644 .9843 


40 


1.3032 


.2705 


30 


.2672 


.4269 


.2773 


.4430 


3.6059 


.5570 .9636 .9839 


30 


1.3003 


.2734 


40 


.2100 


.4314 


.2805 


.4479 


3.5656 


.5521 


.9628 .9836 


20 


1.2974 


.2763 


50 


.2728 


.4359 


.2836 


.4527 


3.5261 


.5473 


.9621 .9832 


10 


1.2945 


.2793 


16^00 


.2756 


.4403 


.2867 


.4575 


3.4874 


.5425 


.9613 .9828 


74° 00' 


1.2915 


.2822 


10 


.2784 


.4447 


.2899 


.4622 


3.4495 


.5378 


.9605 .9825 


50 


1.2886 


.2851 


20 


.2812 


.4491 


.2931 


.46()9 


3.4124 


.5331 .9596 .9821 


40 


1.2857 


.28.80 


30 


.2840 


.4533 


.2962 


.4716 


3.3759 


.5284 


.9588 .9817 


30 


1.2828 


.2^)r)9 


40 


.28(38 


.4576 


.2994 


.4762 


3.3402 


.5238 


.9580 .9814 


20 


1.2799 


.29:38 


50 


.2896 


.4618 


.3026 


.4808 


3.3052 


.5192 


.9572 .9810 


10 


1.2770 


.2967 


17^00' 


.2924 


.4659 


.3057 


.4853 


3.2709 


.5147 


.9563 .9806 


73° 00' 


1.2741 


.29r>6 


10 


.2952 


.4700 


.3089 


.4898 i 3.2371 


.5102 i .9555 .9802 


50 


1.2712 


.3025 


20 


.2979 


.4741 


..3121 


.4943 3.2041 


.5057 1 .9546 .9798 


40 


1.2683 


.3054 


30 


.3007 


.4781 


.3153 


.4987 


3.1716 


.,5013 


.9537 .9794 


30 


1.2654 


.3083 


40 


..3o:r> 


.4821 


.3ia5 


..5031 


's.w.n 


.4969 


.9528 .9790 


20 


1.2625 


.3113 


5i) 


.30()2 


.4861 


.;3217 


.5075 


3.10^ 


.4925 


.9520 .978*3 


10 


1.2595 


.3142 


18^00' 


.3090 


.4900 


.3249 


.5118 


3.0777 


.4882 


.9511 .9782 


72° 00' 


1.2566 






Value 


Lo^io 


Value 


Loerio 


Value 


Loi?io 


Value LojTio 


Degrees 


Radians 






Cosine 


Cotangent 


Tangent 


Sine 







114 Four Place Trigonometric Functions 

[Characteristics of Logarithms omitted — determine by the usual rule from the value] 



Radtans 


Deq-eees 


Sine 


Tangent 


Cotangent | Cosine 






A^VJi.XJ±t*.X^ fj 


j_^ Jl/VTXWJH XjO 


7alue 


Logio 


Value 


Logio 


Value 


Logio 


Value 


Logio 






.3142 


18° 00' 


.3090 


.4900 


.3249 


.5118 


3.0777 


.4882 


.9511 


.9782 


72° 00' 


1.2566 


.3171 


10 


.3118 


.4939 


.3281 


.5161 


3.0475 


.4839 


.9502 


.9778 


50 


1.2537 


.3200 


20 


.3145 


.4977 


.3314 


.5203 


3.0178 


.4797 


.9492 


.9774 


40 


1.2508 


.3229 


30 


.3173 


.5015 


.3346 


.5245 


2.9887 


.4755 


.9483 


.9770 


30 


1.2479 


.3258 


40 


.3201 


.5052 


.3378 


.5287 


2.9600 


.4713 


.9474 


.9765 


20 


1.2450 


.3287 


50 


.3228 


.5090 


.3411 


.5329 


2.9319 


.4671 


.9465 


.9761 


10 


1.2421 


.3316 


19^00' 


.3256 


.5126 


.3443 


.5370 


2.9042 


.4630 


.9455 


.9757 


71° 00' 


1.2392 


,3345 


10 


.3283 


.5163 


.3476 


.5411 


2.8770 


.4589 


.9446 


.9752 


60 


1.2363 


.3374 


20 


.3311 


.5199 


.3508 


.5451 


2.8502 


.4549 


.9436 


.9748 


40 


1.2334 


.3403 


30 


.3338 


.5235 


.3541 


.5491 


2.8239 


.4509 


.9426 


.9743 


30 


1.2305 


.3432 


40 


.3365 


.5270 


.3574 


.5531 


2.7980 


.4469 


.9417 


.9739 


20 


1.2275 


.3462 


50 


.3393 


.5306 


.3607 


.5571 


2.7725 


.4429 


.9407 


.9734 


10 


1.2246 


.3491 


20° 00' 


.3420 


.5341 


.3640 


.5611 


2.7475 


.4389 


.9397 


.9730 


70° 00' 


1.2217 


.3520 


10 


.3448 


.5375 


.3673 


.5650 


2.7228 


.4350 


.9387 


.9725 


50 


1.2188 


.3549 


20 


.3475 


.5409 


.3706 


.5689 


2.6985 


.4311 


.9377 


.9721 


40 


1.2159 


.3578 


30 


.3502 


.5443 


.3739 


.5727 


2.6746 


.4273 


.9367 


.9716 


30 


1.2130 


.3607 


40 


.3529 


.5477 


.3772 


.5766 


2.6511 


.4234 


.9356 


.9711 


20 


1.2101 


.3636 


50 


.3557 


.5510 


.3805 


.5804 


2.6279 


.4196 


.9346 


.9706 


10 


1.2072 


.3665 


21° 00' 


.3584 


.5543 


.3839 


.5842 


2.6051 


.4158 


.9336 


,9702 


69° 00' 


1.2043 


.3694 


10 


.3611 


.5576 


.3872 


.5879 


2.5826 


.4121 


.9325 


.9697 


50 


1.2014 


.3723 


20 


.3638 


.5609 


.3906 


.5917 


2.5605 


.4083 


.9315 


.9692 


40 


1.1985 


.3752 


30 


.3665 


.5641 


.3939 


.5954 


2.5386 


.4046 


.9304 


.9687 


30 


1.1956 


.3782 


40 


.3692 


.5673 


.3973 


.5991 


2.5172 


.4009 


.9293 


.9682 


20 


1.1926 


.3811 


50 


.3719 


.5704 


.4006 


.6028 


2.4960 


.3972 


.9283 


.9677 


10 


1.1897 


.3840 


22° 00' 


.3746 


.5736 


.4040 


.6064 


2.4751 


.3936 


.9272 


.9672 


68° 00' 


1.1868 


.3869 


10 


.3773 


.5767 


.4074 


.6100 


2.4545 


.3900 


.9261 


.9667 


50 


1.1839 


.3898 


20 


.3800 


.5798 


.4108 


.6136 


2.4342 


.3864 


.9250 


.9661 


40 


1.1810 


.3927 


30 


.3827 


.5828 


.4142 


.6172 


2.4142 


.3828 


.9239 


.9656 


30 


1.1781 


.3956 


40 


.3854 


.5859 


.4176 


.6208 


2.3945 


.3792 


.9228 


.9651 


20 


1.1752 


.3985 


50 


.3881 


.5889 


.4210 


.6243 


2.3750 


.3757 


.9216 


.9646 


10 


1.1723 


.4014 


23° 00' 


.3907 


.5919 


.4245 


.6279 


2.3559 


.3721 


.9205 


.9640 


67° 00' 


1.1694 


.4043 


10 


.3934 


.5948 


.4279 


.6314 


2.3369 


.3686 


.9194 


.9635 


50 


1.1665 


.4072 


20 


.3961 


.5978 


.4314 


.6348 


2,3183 


.3652 


.9182 


.9629 


40 


1.1636 


.4102 


30 


.3987 


.6007 


.4348 


.6383 


2.2998 


.3617 


.9171 


.9624 


30 


1.1606 


.4131 


40 


.4014 


.6036 


.4383 


.6417 


2.2817 


.3583 


.9159 


.9618 


20 


1.1577 


.4160 


50 


.4041 


.6065 


.4417 


.6452 


2.2637 


.3548 


.9147 


.9613 


10 


1.1548 


.4189 


24° 00' 


.4067 


.6093 


.4452 


.6486 


2.2460 


.3514 


.9135 


.9607 


66° 00' 


1.1519 


.4218 


10 


.4094 


.6121 


.4487 


.6520 


2.2286 


.3480 


.9124 


.9602 


50 


1.1490 


.4247 


20 


.4120 


.6149 


.4522 


.6553 


2.2113 


.3447 


.9112 


.9596 


40 


1.1461 


.4276 


30 


.4147 


.6177 


.4557 


.6587 


2.1943 


.3413 


.9100 


.9590 


30 


1.1432 


.4305 


40 


.4173 


.6205 


.4592 


.6620 


2.1775 


.3380 


.9088 


.9584 


20 


1.1403 


.4334 


50 


.4200 


.6232 


.4628 


.6654 


2.1609 


.3346 


.9075 


.9579 


10 


1.1374 


.4363 


25° 00' 


.4226 


.6259 


.4663 


.6687 


2.1445 


.3313 


.9063 


.9573 


65° 00' 


1.1345 


.4392 


10 


.4253 


.6286 


.4699 


.6720 


2.1283 


.3280 


.9051 


.9567 


50 


1.1316 


.4422 


20 


.4279 


.6313 


.4734 


.6752 


2.1123 


.3248 


.9038 


.9561 


40 


1.1286 


.4451 


30 


.4305 


.6340 


.4770 


.6785 


2.0965 


.3215 


.9026 


.9555 


30 


1.1257 


.4480 


40 


.4331 


.6366 


.4806 


.6817 


2.0809 


.3183 


.9013 


.9549 


20 


1.1228 


.4509 


50 


.4358 


.6392 


.4841 


.6850 


2.0655 


.3150 


.9001 


.9543 


10 


1.1199 


.4538 


26° 00' 


.4384 


.6418 


.4877 


.6882 


2.0503 


.3118 


.8988 


.9537 


64° 00' 


1.1170 


.4567 


10 


.4410 


.6444 


.4913 


.6914 


2.0353 


.3086 


.8975 


.9530 


50 


1.1141 


.4596 


20 


.4436 


.6470 


.4950 


.6946 


2.0204 


.3054 


.8962 


.9524 


40 


1.1112 


.4625 


30 


.4462 


.6495 


.4986 


.6977 


2.0057 


.3023 


.8949 


.9518 


30 


1.1083 


.4654 


40 


.4488 


.6521 


.5022 


.7009 


1.9912 


.2991 


.8936 


.9512 


20 


1.1054 


.4683 


50 


.4514 


.6546 


.5059 


.7040 


1.9768 


.2960 


.8923 


.9505 


10 


1.1025 


.4712 


27° 00' 


.4540 


.6570 


.5095 


.7072 


1.9626 


.2928 


.8910 


.9499 


63° 00' 


1.0996 






Value 


Logio 


Value 


Logio 


Value 


Logio 


Value 


Logio 


Degrees 


Radians 






Cosine 


Cotangent 


Tangent 


Sine | 







Four Place Trigonometric Functions 



115 



[Characteristics of Logarithms omitted — determine by the usual rule from the value] 


Padiavs 


DFaRFKS 


Sine 


Tange>-t 1 Cotangent | Cosine 








i^i:.(jrax.li>9 


Value 


Logio 


Value 


Logio Value Logir. \'alue Logio 






.4712 


27° 00 


.4540 


.6570 


.5095 


.7072 1.9626 .2928 


.8910 .9499 


63° 00' 


1.0996 


.4741 


10 


.4566 


.6595 


.5i:)2 


.7103 1.94S6 .2897 


.8897 .9492 


50 


1.0966 


.4771 


20 


.4592 


.6620 


.5169 


.7134 


1.9347 .2866 


.8884 .9486 


40 


1.0937 


.4800 


30 


.4617 


.6644 


.5206 


.7165 


1.9210 .2835 


.8870 .9479 


30 


1.0908 


.4829 


40 


.4643 


.66i\S 


.5243 


.7196 


1.9074 .2804 


.8857 .947:^ 


20 


1.0879 


.4858 


50 


.4669 


.6692 


.5280 


.7226 


1.8940 .2774 


.8843 .9466 


10 


1.0850 


.4887 


28^00' 


.4695 


.6716 


.5317 


.7257 


1.8807 .2743 


.8829 .9459 


62° 00 


1.0821 


.4016 


10 


.4720 


.6740 


.5354 


.7287 


1.8676 .2713 


.8816 .94.53 


50 


1.0792 


.4945 


20 


.4746 


.6763 


.5392 


.7317 \ 1.8546 .2683 


.8802 .9446 


40 


1.0703 


.4974 


30 


.4772 


.6787 


.5430 


.7348 1.8418 .2652 


.8788 .94:39 


30 


1.0734 


.5003 


40 


.4797 


.6810 


.5467 


.7378 


1.8291 .2622 


.8774 .94-32 


20 


1.0705 


.5032 


50 


.4823 


.6833 


.5505 


.7408 


1.8165 .2592 


.8760 .9425 


10 


1.0676 


.5061 


29° 00' 


.4848 


.6856 


.5543 


.7438 


1.8040 .2562 


.8746 .9418 


61° 00' 


1.0647 


.5091 


10 


.4874 


.6878 


.5581 


.7467 1.7917 .2533 ! .8752 .9411 


50 


1.0617 


.5120 


20 


.4899 


.6901 


.5619 


.7497 1.7796 .2503 ' .8718 .9404 


40 


1.0.588 


.5149 


30 


.4924 


.6923 


.5658 


.7526 


1.7675 .2474 .8704 .9397 


30 


1.0559 


.5178 


40 


.4950 


.6946 


.5696 


.7556 


1.75.56 .2444 .8(389 .9.390 


20 


1.05.30 


.5207 


50 


.4975 


.6968 


.5735 


.7585 


1.7437 .2415 .8675 .9383 


10 


1.0501 


.5236 


30° 00' 


.5000 


mm 


.5774 


.7614 


1.7321 .2386 .8660 .9375 


60° 00' 


1.0472 


.5265 


10 


.5025 


.7012 


.5812 


.7644 


1.7205 .2356 .8646 .9368 


50 


1.0443 


.5294 


20 


.5050 


.7033 


.5851 


.7673 


1.7090 .2327 ■ .8631 .9361 


40 


1.0414 


.5323 


30 


.5075 


.7055 


.5890 


.7701 


1.6977 .2299 : .8616 .9353 


30 


1.0385 


.5352 


40 


.5100 


.7076 


.5930 


.7730 1.6864 .2270 ' .8601 .9346 


20 


1.0356 


.5381 


50 


.5125 


.7097 


.5969 


.7759 


1.6753 .2241 .8587 .93-38 


10 


1.0327 


.5411 


31°00' 


.5150 


.7118 


.6009 


.7788 


1.6643 .2212 .8572 .9331 


59° 00' 


1.0297 


.5440 


10 


.5175 


.7139 


.6048 


.7816 


1.6534 .2184 .8557 .9323 


50 


1.0268 


.5469 


20 


.5200 


.7160 


.6088 


.7845 1.6426 .2155 , .8542 .9315 


40 


1.02:39 


.5498 


30 


.5225 


.7181 


.6128 


.7873 i 1.6319 .2127 : .8526 .9.308 


30 


1.0210 


.5527 


40 


.5250 


.7201 


.6168 


.7902 


1.6212 .2098 ' .8511 .9300 


20 


1.0181 


.5556 


50 


.5275 


.7222 


.6208 


.7930 


1.6107 .2070 .8496 .9292 


10 


1.0152 


.5585 


32° 00' 


.5299 


.7242 


.6249 


.7958 


1.6003 .2042 .8480 .9284 


58° 00' 


1.0123 


.5614 


10 


.5324 


.7262 


.6289 


.7986 


1.5900 .2014 \ .8465 .9276 


50 


1.0094 


.5643 


20 


.5:348 


.7282 


.6330 


.8014 


1.5798 .1986 i .8450 .9268 


40 


1.0065 


.5672 


30 


.5373 


.7302 


.6371 


.8042 


1.5697 .1958 .84-34 .9260 


30 


1.003() 


.5701 


40 


.5398 


.7322 


.6412 


.8070 


1.5597 .1930 .8418 .9252 


20 


1.0007 


.5730 


50 


.5422 


.7342 


.6453 


.8097 


1.5497 .1903 1.8403 .9244 


10 


.9977 


.5760 


33° 00' 


.5446 


.7361 


.6494 


.8125 


1.5-399 .1875 .8387 .92-36 


57°00' 


.9948 


.5789 


10 


.5471 


.7380 


.6536 


.8153 


1.5:301 .1847 .8-371 .9228 


50 


.9919 


.5818 


20 


.5495 


.7400 


.6577 


.8180 


1.5204 .1820 .8:355 .9219 


40 


.9890 


.5847 


30 


.5519 


.7419 


.6619 


.8208 


1.5108 .1792 .83:39 .9211 


30 


.9861 


.587() 


40 


.5544 


.7438 1 .6661 


.8235 


1.5013 .1765 .8-323 .9203 


20 


.98-32 


.5905 


50 


.5568 


.7457 


.6703 


.8263 


1.4919 .1737 .8307 .9194 


10 


.9803 


.5934 


34° 00' 


.5592 


.7476 


.6745 


.8290 


1.4826 .1710 .8290 .9186 


56° 00' 


.9774 


.5963 


10 


.5616 


.7494 


.6787 


.8317 


1.4733 .1683 | .8274 .9177 


50 


.9745 


.5992 


20 


.5640 


.7513 


.6830 


.8344 


1.4641 .1656! .8258' .9169 


40 


.9716 


.6021 


30 


.5604 


.7531 


.6873 


.8371 ! 1.4550 .1629 i .8241 .9160 


30 


.9687 


.6050 


40 


.5688 


.7550 


.6916 


.8398 1.4460 .1602 .8225 .9151 


20 


.9()57 


.6080 


50 


.5712 


.7568 


.6959 


.8425 1.4370 .1575 j .8208 .9142 


10 


.9628 


.6109 


35° 00' 


.5736 


.7586 


.7002 


.8452 1.4281 .1548 ' .8192 .9134 


55° 00' 


.9599 


.6138 


10 


.5760 


.7604 


.7046 


.8479 1.4193 .1521 .8175 .9125 


50 


.9570 


.6167 


20 


.5783 


.7622 


.7089 


.85(J6 1 1.4106 .1494 .8158 .9116 


40 


.9541 


.6196 


30 


.5807 


.7640 


.7133 


.8533 


1.4019 .1467 .8141 .9107 


30 


.9512 


.6225 


40 


.5831 


.7657 


.7177 


.8.559 


1.3934 .1441 


.8124 .9098 


20 


.9483 


.6254 


50 


.5854 


.7675 


.7221 


.8586 


1.3^8 .1414 


.8107 .9089 


10 


.9454 


.6283 


36° 00' 


.5878 


.7692 


.7265 


.8613 ' 1.3764 .1387 ' .8090 .9080 


54° 00' 


.9425 






Value 


Logio 


Value 


LoETio' Value Logjo! Value Logio 


Degbees 


Radians 






COR'INE 


Cotangent i Tangent ( Sine 







116 Four Place Trigonometric Functions 

[Characteristics of Logarithms omitted — determine by the usual rule from the value] 



"Radians 


DEG-RKEft 


Sine 


Tangent 


Cotangent 


Cosine 






■LWXVA'X.clb.X^ O 


JU' .Cl VT X« JU •>» E7 


7alue Logic 


Value Logic 


Value Logic 


Value Logic 






.6283 


36^^00' 


.5878 .7682 


.7265 .8613 


1.3764 .1387 


.8090 .9080 


64° 00' 


.9425 


.6312 


10 


.5901 .7710 


.7310 .8639 


1.3680 .13(31 


.8073 .9070 


50 


.9396 


.6341 


20 


.5925 .7727 


.7355 .8666 


1.3597 .1334 


.8056 .9061 


40 


.9367 


.6370 


30 


.5948 .7744 


.7400 .8692 


1.3514 .1308 


.8039 .9052 


30 


.9338 


.6400 


40 


.5972 .7761 


.7445 .8718 


1.3432 .1282 


.8021 .9042 


20 


.9308 


.6429 


50 


.5995 .7778 


.7490 .8745 


1.3351 .1255 


.8004 .9033 


10 


.9279 


.6458 


37° 00' 


.6018 .7795 


.7536 .8771 


1.3270 .1229 


.7986 .9023 


63° 00' 


.9250 


.6487 


10 


.6041 .7811 


.7581 .8797 


1.3190 .1203 


.7969 .9014 


50 


.9221 


.6516 


20 


.6065 .7828 


.7627 .8824 


1.3111 .1176 


.7951 .9004 


40 


.9192 


.6545 


30 


.6088 .7844 


.7673 .8850 


1.3032 .1150 


.7934 .8995 


30 


.9163 


.6574 


40 


.6111 .7861 


.7720 .8876 


1.2954 .1124 


.7916 .8985 


20 


.9134 


.6603 


50 


.6134 .7877 


.7766 .8902 


1.2876 .1098 


.7898 .8975 


10 


.9105 


.6632 


38° 00' 


.6157 .7893 


.7813 .8928 


1.2799 .1072 


.7880 .8965 


52° 00' 


.9076 


.6661 


10 


.6180 .7910 


.7860 .8954 


1.2723 .1046 


.7862 .8955 


50 


.9047 


.6690 


20 


.6202 .7926 


.7907 .8980 


1.2647 .1020 


.7844 .8945 


40 


.9018 


.6720 


30 


.6225 .7941 


.7954 .9006 


1.2572 .0994 


.7826 .8935 


30 


.8988 


.6749 


40 


.6248 .7957 


.8002 .V)032 


1.2497 .0968 


.7808 .8925 


20 


.8959 


.6778 


50 


.6271 .7973 


.8050 .9058 


1.2423 .0942 


.7790 .8915 


10 


.8930 


.6807 


39° 00' 


.6293 .7989 


.8098 .9084 


1.2349 .0916 


.7771 .8905 


51°00' 


.8901 


.6836 


10 


.6316 .8004 


.8146 .9110 


1.2276 .0890 


.7753 .8895 


50 


.8872 


.6865 


20 


.6338 .8020 


.8195 .9135 


1.2203 .0865 


.7735 .8884 


40 


.8843 


.6894 


30 


.6361 .8035 


.8243 .9161 


1.2131 .0839 


.7716 .8874 


30 


.8814 


.6923 


40 


.6383 .8050 


.8292 .9187 


1.2059 .0813 


.7698 .8864 


20 


.8785 


.6952 


50 


.6406 .8066 


.8342 .9212 


1.1988 .0788 


.7679 .8853 


10 


.8756 


.6981 


40° 00' 


.6428 .8081 


.8391 .9238 


1.1918 .0762 


.7660 .8843 


50° 00' 


.8727 


.7010 


10 


.6450 .8096 


.8441 .9264 


1.1847 .0736 


.7642 .8832 


50 


.8698 


.7039 


20 


.6472 .8111 


.8491 .9289 


1.1778 .0711 


.7623 .8821 


40 


.8668 


.7069 


30 


.6494 .8125 


.8541 .9315 


1.1708 .0685 


.7604 .8810 


30 


.8639 


.7098 


40 


.6517 .8140 


.8591 .9341 


1.1640 .0659 


.7585 .8800 


20 


.8610 


.7127 


50 


.6539 .8155 


.8642 .9366 


1.1571 .0634 


.7566 .8789 


10 


.8581 


.7156 


41° 00' 


.6561 .8169 


.8693 .9392 


1.1504 .0(^8 


.7547 .8778 


49° 00' 


.8552 


.7185 


10 


.6583 .8184 


.8744 .9417 


1.1436 .0583 


.7528 .8767 


50 


.8523 


.7214 


20 


.6604 .8198 


.8796 .9443 


1.1369 .0557 


.7509 .8756 


40 


.8494 


.7243 


30 


.6626 .8213 


.8847 .9468 


1.1303 .0532 


.7490 .8745 


30 


.8465 


.7272 


40 


.6648 .8227 


.8899 .9494 


1.1237 .0506 


.7470 .8733 


20 


.8436 


.7301 


50 


.6670 .8241 


.8952 .9519 


1.1171 .0481 


.7451 .8722 


10 


.8407 


.7330 


42° 00 


.6691 .8255 


.9004 .9544 


1.1106 .0456 


.7431 .8711 


48° 00' 


.8378 


.7359 


10 


.6713 .8269 


.9057 .9570 


1.1041 .0430 


.7412 .8699 


50 


.8348 


.7389 


20 


.6734 .8283 


.9110 .9595 


1.0977 .0405 


.7392 .8688 


40 


.8319 


.7418 


30 


.6756 .8297 


.9163 .9621 


1.0913 .0379 


.7373 .8676 


30 


.8290 


.7447 


40 


.6777 .8311 


.9217 .9646 


1.0850 .0354 


.7353 .8665 


20 


.8261 


.7476 


50 


.6799 .8324 


.9271 .9671 


1.0786 .0329 


.7333 .8653 


10 


.8232 


.7505 


43° 00' 


.6820 .8338 


.9325 .9697 


1.0724 .0303 


.7314 .8641 


47° 00' 


.8203 


.7534 


10 


.6841 .8351 


.9380 .9722 


1.0661 .0278 


.7294 .8629 


50 


.8174 


.7563 


20 


.6862 .8365 


.9435 .9747 


1.0599 .0253 


.7274 .8618 


40 


.8145 


.7592 


30 


.6884 .8378 


.9490 .9772 


1.0538 .0228 


.7254 .8606 


30 


.8116 


.7621 


40 


.6905 .8391 


.9545 .9798 


1.0477 .0202 


.7234 .8594 


20 


.8087 


.7650 


50 


.6926 .8405 


.9601 .9823 


1.0416 .0177 


.7214 .8582 


10 


.8058 


.7679 


44° 00' 


.6947 .8418 


.9657 .9848 


1.0355 .0152 


.7193 .8569 


46° 00' 


.8029 


.7709 


10 


.6967 .8431 


.9713 .9874 


1.0295 .0126 


.7173 .8557 


50 


.7999 


.7738 


20 


.6988 .8444 


.9770 .9899 


1.0235 .0101 


.7153 .8545 


40 


.7970 


.7767 


30 


.7009 .8457 


.9827 .9924 


1.0176 .0076 


.7133 .8532 


30 


.7941 


.7796 


40 


.7030 .8469 


.9884 .9949 


1.0117 .0051 


.7112 .8520 


20 


.7912 


.7825 


50 


.7050 .8482 


.9942 .9975 


1.0058 .0025 


.7092 .8507 


10 


.7883 


.7854 


45° 00' 


.7071 .8495 


1.0000 .0000 


1.0000 .0000 


.7071 .8495 


45° 00' 


.7854 






Value Logio 


Value Logio 


Value Logio 


Value Logio 


Degrees 


Jadians 






Cosine 


Cotangent 


Tangent 


Sine 







Values aud Logtirithius of Haversines 



117 





[Characteristics ^ 


:>{ Logarithms omitted - 


— determine by rule from the value] 






0' 




10' 1 


20' 


3C 


' 


40' 


oC 


)' 




Value 


Logio 


\alue 


I^ogio 


Value 


Log,o 


Value 


Logio 


Value 


Log,o 


Value 


Logjo 





.0000 




.0000 4.3254 


.0000 4.9275 


.0000 t 


).2796 


.0000 5.5295 


.0001 5.7233 1 


1 


.0001 5.8817 


.0001 6.0156 


.0001 6.1315 


.0002 


.2338 


.0002 


.3254 


.0003 


.4081 


2 


.0003 


.4837 


.0004 


.5532 


.0004 


.6176 


.0005 


.6775 


.0005 


.7336 


.0006 


.7862 


3 


.0007 


.8358 


.0008 


.8828 


.0008 


.9273 


.0009 


.9697 


.0010 


.0101 


.0011 


.0487 


4 


.0012 


.0856 


.0013 


.1211 


.0014 


.1551 


.0015 


.1879 


.0017 


.2195 


.0018 


.2499 


5 


.0019 


.2793 


.0020 


.3078 


.0022 


.3354 


.0023 


.3621 


.0024 


.3880 


.0026 


.4132 


6 


.0027 


.4376 


.0029 


.4614 


.0031 


.4845 


.0032 


.5071 


.0034 


.5290 


.0036 


.5504 


7 


.0037 


.5713 


.0039 


.5918 


.0041 


.6117 


.0043 


.6312 


.0045 


.6503 


.0047 


.6689 


S 


.0049 


.6872 


.0051 


.7051 


.0053 


.7226 


.0055 


.7397 


.0057 


.7566 


.0059 


.7731 


9 


.0062 


.7893 


.0064 


.8052 


.0066 


.8208 


.0069 


.8361 


.0071 


.8512 


.0073 


.8660 


10 


.0076 


.8806 


.0079 


.8949 


.0081 


.9090 


.0084 


.9229 


.0086 


.9365 


.0089 


.9499 


11 


.0092 


.9631 


.0095 


.9762 


.0097 


.9^90 


.0100 


.0016 


.0103 


.0141 


.0106 


.0264 


12 


.0109 


.0385 


.0112 


.0504 


.0115 


.0622 


.0119 


.073b 


.0122 


.0853 


.0125 


.0966 


13 


.0128 


.1077 


.0131 


.1187 


.0135 


.1296 


.0138 


.1404 


.0142 


.1510 


.0145 


.1614 


U 


.0149 


.1718 


.0152 


.1820 


.0156 


.1921 


.0159 


.2021 


.0163 


.2120 


.0167 


.2218 


15 


.0170 


.2314 


.0174 


.2409 


.0178 


.2504 


.0182 


.2597 


.0186 


.2689 


.0190 


.2781 


16 


.0194 


.2871 


.0198 


.2961 


.0202 


.3049 


.0206 


.3137 


.0210 


.3223 


.0214 


.3309 


17 


.0218 


.3394 


.0223 


.3478 


.0227 


.3561 


.0231 


.3644 


.0236 


.3726 


.0240 


.3806 


IS 


.0245 


.3887 


.0249 


.3966 


.0254 


.4045 


.0258 


.4123 


.0263 


.4200 


.0268 


.4276 


19 


.0272 


.4352 


.0277 


.4427 


.0282 


.4502 


.0287 


.4576 


.0292 


.4649 


.0297 


.4721 


20 


.0302 


.4793 


.0307 


.4865 


.0312 


.4936 


.0317 


.5006 


.0322 


.5075 


.0327 


.5144 


21 


.0332 


.5213 


.0337 


.5281 


.0343 


.5348 


.0348 


.5415 


.0353 


.5481 


.0359 


.5547 


22 


.0364 


.5612 


.0370 


.5677 


.0375 


.5741 


.0381 


.5805 


.0386 


.5868 


.0392 


.5931 


23 


.0397 


.5993 


.0403 


.6055 


.0409 


.6116 


.0415 


.6177 


.0421 


.6238 


.0426 


.6298 


24 


.0432 


.6357 


.0438 


.6417 


.0444 


.6476 


.0450 


.6534 


.0456 


.6592 


.0462 


.6650 


25 


.0468 


.6707 


.0475 


.6764 


.0481 


.6820 


.0487 


.6876 


.0493 


.6932 


.0500 


.6987 


26 


.0506 


.7042 


.0512 


.7096 


.0519 


.7151 


.0525 


.7204 


.0532 


.7258 


.0538 


.7311 


27 


.0545 


.7364 


.0552 


.7416 


.0558 


.7468 


.0565 


.7520 


.0572 


.7572 


.0578 


.7623 


28 


.0585 


.7673 


.0592 


.7724 


.0599 


.7774 


.0606 


.7824 


.0613 


.7874 


.0620 


.7923 


29 


.0627 


.7972 


.0634 


.8020 


.0641 


.8069 


.0648 


.8117 


.0655 


.8165 


.0663 


.8213 


30 


.0670 


.8260 


.0677 


.8307 


.0684 


.8354 


.0692 


.8400 


.0699 


.8446 


.0707 


.8492 


31 


.0714 


.8538 


.0722 


.8583 


.0729 


.8629 


.0737 


.8673 


.0744 


.8718 


.0752 


.8763 


32 


.0760 


.8807 


.0767 


.8851 


.0775 


.8894 


.0783 


.8938 


.0791 


.8981 


.0799 


.9024 


33 


.0807 


.9067 


.0815 


.9109 


.0823 


.9152 


.0831 


.9194 


.0839 


.9236 


.0847 


.9277 


34 


.0855 


.9319 


.0863 


.9360 


.0871 


.9401 


.0879 


.9442 


.0888 


.9482 


.0896 


.9523 


35 


.0904 


.9563 


.0913 


.9603 


.0921 


.9643 


.0929 


.9682 


.0938 


.9722 


.0946 


.9761 


36 


.0955 


.9800 


.0963 


.9838 


.0972 


.9877 


.0981 


.9915 


.0989 


.9954 


.0998 


.9992 


37 


.1007 


.0030 


.1016 


.0067 


.1024 


.0105 


.1033 


.0142 


.1042 


.0179 


.1051 


.0216 


38 


.1060 


.0253 


.1069 


.0289 


.1078 


.0326 


.1087 


.0362 


.1096 


.0398 


.1105 


.0434 


39 


.1114 


.0470 


.1123 


.0505 


.1133 


.0541 


.1142 


.0576 


.1151 


.0611 


.1160 


.0646 


40 


.1170 


.0681 


.1179 


.0716 


.1189 


.0750 


.1198 


.0784 


.1207 


.0817 


.1217 


.0853 


41 


.1226 


.08b7 


.1236 


.0920 


.1246 


.0954 


.1255 


.0987 


.1265 


.1021 


.1275 


.1054 


42 


.1284 


.1087 


.1294 


.1119 


.1304 


.1152 


.1314 


.1185 


.1323 


.1217 


.1333 


.1249 


43 


.1343 


.1282 


.1353 


.1314 


.1363 


.1345 


.1373 


.1377 


.1383 


.1409 


.1393 


.1440 


44 


.1403 


.1472 


.1413 


.1503 


.1424 


.1534 


.1434 


.1565 


.1444 


.1596 


.1454 


.1626 


45 


.1464 


.1657 


.1475 


.1687 


.1485 


.1718 


.1495 


.1748 


.1506 


.1778 


.1516 


.1808 


46 


.1527 


.1838 


.1538 


.1867 


.1548 


.1897 


.1558 


.1926 


.1569 


.1956 


.1579 


.1985 


47 


.1590 


.2014 


.1600 


.2043 


.1611 


.2072 


.1622 


.2101 


.1633 


.2129 


.1644 


.2158 


48 


.1654 


.2186 


.1665 


.2215 


.1676 


.2243 


.1687 


.2271 


.1698 


.2299 


.1709 


.2327 


49 


.1720 


.2355^ 1731 


.2382 


.1742 


.2410 


.1753 


.2437 


.1764 


.2465 


.1775 


.2492 


50 


.1786 


.2519 


.1797 


.2546 


.1808 


.2573 


.1820 


.2600 


.1831 


.2627 


.1842 


.2653 


51 


.1853 


.2680 


.1S65 


.2706 


.1876 


.2732 


.1887 


.2759 


.1899 


.2785 


.1910 


.2811 


52 


.1922 


.2837 


.1933 


.2863 


.1945 


.2888 


.1956 


.2914 


.1968 


.2940 


.1979 


.2965 


53 


.1991 


.2991 


.2003 


.3016 


.2014 


.3041 


.2026 


.3066 


.2038 


.3091 


.2049 


.3116 


54 


.2061 


.3141 


.2073 


.3166 


.2085 


.3190 


.2096 


.3215 


.2108 


.3239 


.2120 


.3264 


55 


.2132 


.3288 


' .2144 


.3312 


.2156 


.3336 


.2168 


.3361 


.2180 


.3384 


.2192 


.3408 


56 


.2204 


.3432 


.2216 


.3456 


.2228 


.3480 


.2240 


.3503 


.2252 


.3527 


.2265 


.3550 


57 


.2277 


.3573 


.2289 


.3596 


.2301 


.3620 


.2314 


.3643 


.2326 


.3666 


.2338 


.3689 


58 


.2350 


.3711 


.2363 


.3734 


.2375 


.3757 


.2388 


.3779 


.2400 


.3802 


.2412 


.3824 


59 


.2425 


.3847 


.2437 


.3869 


.2450 


.3891 


.2462 


.3913 


.2475 


.3935 


.2487 


.3957 



118 Values and Logarithms of Haversines 

[Characteristics of Logarithms omitted — determine by rule from the value] 



o 





' 


10' 


20' 


30' 


40' 


50' 




Value 


Logjo 


Value 


Logio 


Value 


Logio 


Value 


Logio 


Value 


Logio 


Value 


I^ogio 


60 


.2500 


.3979 


.2513 


.4001 


.2525 


.4023 


.2538 


.4045 


.2551 


.4066 


.2563 


.4088 


61 


.2576 


.4109 


.2589 


.4131 


.2601 


.4152 


.2614 


.4173 


.2627 


.4195 


.2640 


.4216 


62 


.2653 


.4237 


.2665 


.4258 


.2678 


.4279 


.2691 


.4300 


.2704 


.4320 


.2717 


.4341 


63 


.2730 


.4362 


.2743 


.4382 


.2756 


.4403 


.2769 


.4423 


.2782 


.4444 


.2795 


.4464 


64 


.2808 


.4484 


.2821 


.4504 


.2834 


.4524 


.2847 


.4545 


.2861 


.4565 


.2874 


.4584 


65 


.2887 


.4604 


,2900 


.4624 


.2913 


.4644 


.2927 


.4664 


.2940 


.4683 


.2953 


.4703 


66 


.2966 


.4722 


.2980 


.4742 


.2993 


.4761 


.3006 


.4780 


.3020 


.4799 


.3033 


.4819 


67 


.3046 


.4838 


.3060 


.4857 


.3073 


.4876 


.3087 


.4895 


.3100 


.4914 


.3113 


.4932 


68 


.3127 


.4951 


.3140 


.4970 


.3154 


.4989 


.3167 


.5007 


.3181 


.5026 


.3195 


.5044 


69 


.3208 


.5063 


.3222 


.5081 


.3235 


.5099 


.3249 


.5117 


.3263 


.5136 


.3276 


.5154 


70 


.3290 


.5172 


.3304 


.5190 


.3317 


.5208 


.3331 


.5226 


.3345 


.5244 


.3358 


.5261 


71 


.3372 


.5279 


.3386 


.5297 


.3400 


.5314 


.3413 


.5332 


.3427 


.5349 


.3441 


.5367 


72 


.3455 


.5384 


.3469 


.5402 


.3483 


.5419 


.3496 


.5436 


.3510 


.5454 


.3524 


.5471 


73 


.3538 


.5488 


.3552 


.5505 


.3566 


.5522 


.3580 


.5539 


.3594 


.5556 


.3608 


.5572 


74 


.3622 


.5589 


.3636 


.5606 


.3650 


.5623 


.3664 


.5639 


.3678 


.5656 


.3692 


.5672 


75 


.3706 


.5689 


.3720 


.5705 


.3734 


.5722 


.3748 


.5738 


.3762 


.5754 


.3776 


.5771 


76 


.3790 


.5787 


.3805 


.5803 


.3819 


.5819 


.3833 


.5835 


.3847 


.5851 


.3861 


.5867 


77 


.3875 


.5883 


.3889 


.5899 


.3904 


.5915 


.3918 


.5930 


.3932 


.5946 


.3946 


.5962 


78 


.3960 


.5977 


.3975 


.5993 


.3989 


.6009 


.4003 


.6024 


.4017 


.6039 


.4032 


.6055 


79 


.4046 


.6070 


.4060 


.6085 


.4075 


.6101 


.4089 


.6116 


.4103 


.6131 


.4117 


.6146 


80 


.4132 


.6161 


.4146 


.6176 


.4160 


.6191 


.4175 


.6206 


.4189 


.6221 


.4203 


.6236 


81 


.4218 


.6251 


.4232 


.6266 


.4247 


.6280 


.4261 


.6295 


.4275 


.6310 


.4290 


.6324 


82 


.4304 


.6339 


.4319 


.6353 


.4333 


.6368 


.4347 


.6382 


.4362 


.6397 


.4376 


.6411 


83 


.4391 


.6425 


.4405 


.6440 


.4420 


.6454 


.4434 


.6468 


.4448 


.6482 


.4463 


.6496 


84 


.4477 


.6510 


.4492 


.6524 


.4506 


.6538 


.4521 


.6552 


.4535 


.6566 


.4550 


.6580 


85 


.4564 


.6594 


.4579 


.6607 


.4593 


.6621 


.4608 


.6635 


.4622 


.6649 


.4637 


.6662 


86 


.4651 


.6676 


.4666 


.6689 


.4680 


.6703 


.4695 


.6716 


.4709 


.6730 


.4724 


.6743 


87 


.4738 


.6756 


.4753 


.6770 


.4767 


.6783 


.4782 


.6796 


.4796 


.6809 


.4811 


.6822 


88 


.4826 


.6835 


.4840 


.6848 


.4855 


.6862 


.4869 


.6875 


.4884 


.6887 


.4898 


.6900 


89 


.4913 


.6913 


.4937 


.6926 


.4942 


.6939 


.4956 


.6952 


.4971 


.6964 


.4985 


.6977 


90 


.5000 


.6990 


.5015 


.7002 


.5029 


.7015 


.5044 


.7027 


.5058 


.7040 


.5073 


.7052 


91 


.5087 


.7065 


.5102 


.7077 


.5116 


.7090 


.5131 


.7102 


.5145 


.7114 


.5160 


.7126 


92 


.5174 


.7139 


.5189 


.7151 


.5204 


.7163 


.5218 


.7175 


.5233 


.7187 


.5247 


.7199 


93 


.5262 


.7211 


.5276 


.7223 


.5291 


.7235 


.5305 


.7247 


.5320 


.7259 


.5334 


.7271 


94 


.5349 


.7283 


.5363 


.7294 


.5378 


.7306 


.5392 


.7318 


.5407 


.7329 


.5421 


.7341 


95 


.5436 


.7353 


.5450 


.7364 


.5465 


.7376 


.5479 


.7387 


.5494 


.7399 


.5508 


.7410 


96 


.5523 


.7421 


.5537 


.7433 


.5552 


.7444 


.5566 


.7455 


.5580 


.7467 


.5595 


.7478 


97 


.5609 


.7489- 


.5624 


.7500 


.5638 


.7511 


.5653 


.7523 


.5667 


.7534 


.5682 


.7545 


98 


.5696 


.7556 


.5710 


.7567 


.5725 


.7577 


.5739 


.7588 


.5753 


.7599 


.5768 


.7610 


99 


.5782 


.7621 


.5797 


.7632 


.5811 


.7642 


.5825 


.7653 


.5840 


.7664 


.5854 


.7674 


100 


.5868 


.7685 


.5883 


.7696 


.5897 


.7706 


.5911 


.7717 


.5925 


.7727 


.5940 


.7738 


101 


.5954 


.7748 


.5968 


.7759 


.5983 


.7769 


.5997 


.7779 


.6011 


.7790 


.6025 


.7800 


102 


.6040 


.7810 


.6054 


.7820 


.6068 


.7830 


.6082 


.7841 


.6096 


.7851 


.6111 


.7861 


103 


.6125 


.7871 


.6139 


.7881 


.6153 


.7891 


.6167 


.7901 


.6181 


.7911 


.6195 


.7921 


104 


.6210 


.7931 


.6224 


.7940 


.6238 


.7950 


.6252 


.7960 


.6266 


.7970 


.6280 


.7980 


105 


.6294 


.7989 


.6308 


.7999 


.6322 


.8009 


.6336 


.8018 


.6350 


.8028 


.6364 


.8037 


106 


.6378 


.8047 


.6392 


.8056 


.6406 


.8066 


.6420 


.8075 


.6434 


.8085 


.6448 


.8094 


107 


.6462 


.8104 


.6476 


.8113 


.6490 


.8122 


.6504 


.8131 


.6517 


.8141 


.6531 


.8150 


108 


.6545 


.8159 


.6559 


.8168 


.6573 


.8177 


.6587 


.8187 


.6600 


.8196 


.6614 


.8205 


109 


.6628 


.8214 


.6642 


.8223 


.6655 


.8232 


.6669 


.8241 


.6683 


.8250 


.6696 


.8258 


110 


.6710 


.8267 


.6724 


.8276 


.6737 


.8285 


.6751 


.8294 


.6765 


.8302 


.6778 


.8311 


111 


.6792 


.8320 


.6805 


.8329 


.6819 


.8337 


.6833 


.8346 


.6846 


.8354 


.6860 


.8363 


112 


.6873 


.8371 


.6887 


.8380 


.6900 


.8388 


.6913 


.8397 


.6927 


.8405 


.6940 


.8414 


113 


.6954 


.8422 


.6967 


.8430 


.6980 


.8439 


.6994 


.8447 


.7007 


.8455 


.7020 


.8464 


114 


.7034 


.8472 


.7047 


.8480 


.7060 


.8488 


.7073 


.8496 


.7087 


.8504 


.7100 


.8513 


115 


.7113 


.8521 


.7126 


.8529 


.7139 


.8537 


.7153 


.8545 


.7166 


.8553 


.7179 


.8561 


116 


.7192 


.8568 


.7205 


.8576 


.7218 


.8584 


.7231 


.8592 


.7244 


.8600 


.7257 


.8608 


117 


.7270 


.8615 


.7283 


.8623 


.7296 


.8631 


.7309 


.8638 


.7322 


.8646 


.7335 


.8654 


178 


.7347 


.8661 


.7360 


.8669 


.7373 


.8676 


.7386 


.8684 


.7399 


.8691 


.7411 


.8699 


119 


.7424 


.8706 


.7437 


.8714 


.7449 


.8721 


.7462 


.8729 


.7475 


.8736 


.7487 


.8743 



Values jiud Logarithms of Haversines 

[Characteristics of Logarithms omitted — determine by rule from the value] 



119 



^ 


C 




10' 


20' 


30' 


40' 


50' 1 




Value 


I-ogio 


Value 


I>og,o 


Value 


Logio 


Value 


Logio 


Value 


T^ogio 


Value 


Login 


120 


.7500 


.8751 


.7513 


.8758 


.7525 


.8765 


.7538 


.8772 


.7550 


.8780 .7563 


.8787 


121 


.7575 


.8794^ 


.7588 


.8801 


.7600 


.8808 


.7612 


.8815 


.7625 


.8822 


> .7637 


.8829 


122 


.7650 


.8836 


.7662 


.8843 


.7674 


.8850 


.7686 


.8857 


.7699 


.886^ 


.7711 


.8871 


123 


.7723 


.8878 


.7735 


.8885 


.7748 


.8892 


.7760 


.8898 


.7772 


.8905 


.7784 


.8912 


124 


.7796 


.8919 


.7808 


.8925 


.7820 


.8932 


.7832 


.8939 


.7844 


.8945 


.7856 


.8952 


125 


.7868 


.8959 


.7880 


.8965 


.7892 


.8972 


.7904 


.8978 


.7915 


.8985 


.7927 


.8991 


126 


.7939 


.8998 


.7951 


.9004 


.7962 


.9010 


.7974 


.9017 


.7986 


.9023 


.7997 


.9030 


127 


.8009 


.9036 


.8021 


.9042 


.8032 


.9048 


.8044 


.9055 


.8055 


.9061 


.8067 


.9067 


128 


.8078 


.9073 


.8090 


.9079 


.8101 


.9085 


.8113 


.9092 


.8124 


.9098 


.8135 


.9104 


129 


.8147 


.9110 


.8158 


.9116 


.8169 


.9122 


.8180 


.9128 


.8192 


.9134 


.8203 


.9140 


130 


.8214 


.9146 


.8225 


.9151 


.8236 


.9157 


.8247 


.9163 


.8258 


.9169 


.8269 


.9175 


131 


.8280 


.9180 


.8291 


.9186 


.8302 


.9192 


.8313 


.9198 


.8324 


.9203 


.8335 


.9209 


132 


.8346 


.9215 


.8356 


.9220 


.8367 


.9226 


.8378 


.9231 


.8389 


.9237 


.8399 


.9242 


133 


.8410 


.9248 


.8421 


.9253 


.8431 


.9259 


.8442 


.9264 


.8452 


.9270 


.8463 


.9275 


134 


.8473 


.9281 


.8484 


.9286 


.8494 


.9291 


.8501 


.9297 


.8515 


.9302 


.8525 


.9307 


135 


.8536 


.9312 


.8546 


.9318 


.8556 


.9323 


.8566 


.9328 


.8576 


.9333 


.8587 


.9338 


136 


.8597 


.9343 


.8607 


.9348 


.8617 


.9353 


.8627 


.9359 


.8637 


.9364 


.8647 


.9369 


137 


.8657 


.9374 


.8667 


.9379 


.8677 


.9383 


.8686 


.9388 


.8696 


.9393 


~ .8706 


.9398 


138 


.8716 


.9403 


.8725 


.9408 


.8735 


.9413 


.8745 


.9417 


.8754 


.9422 


.8764 


.9427 


139 


.8774 


.9432 


.8783 


.9436 


.8793 


.9441 


.8802 


.9446 


.8811 


.9450 


.8821 


.9455 


140 


.8830 


.9460 


.8840 


.9464 


.8849 


.9469 


.8858 


.9473 


.8867 


.9478 


.8877 


.9482 


141 


.8886 


.9487 


.8895 


.9491 


.8904 


.9496 


.8913 


.9500 


.8922 


.9505 


.8931 


.9509 


142 


.8940 


.9513 


.8949 


.9518 


.8958 


.9522 


.8967 


.9526 


.8976 


.9531 


.8984 


.9535 


143 


.8993 


.9539 


.9002 


.9543 


.9011 


.9548 


.9019 


.9552 


.9028 


.9556 


.9037 


.9560 


144 


.9045 


.9564 


.9054 


.9568 


.9062 


.9572 


.9071 


.9576 


.9079 


.9580 


.9087 


.9584 


145 


.9096 


.9588 


.9104 


.9592 


.9112 


.9596 


.9121 


.9600 


.9129 


.9604 


.9137 


.9608 


146 


.9145 


.9612 


.9153 


.9616 


.9161 


.9620 


.9169 


.9623 


.9177 


.9627 


.9185 


.9631 


147 


.9193 


.9635 


.9201 


.9638 


.9209 


.9642 


.9217 


.9646 


.9225 


.9650 


.9233 


.9653 


148 


.9240 


.9657 


.9248 


.9660 


.9256 


.9664 


.9263 


.9668 


.9271 


.9671 


.9278 


.9675 


149 


.9286 


.9678 


.9293 


.9682 


.9301 


.9685 


.9308 


.9689 


.9316 


.9692 


.9323 


.9695 


150 


.9330 


.9699 


.9337 


.9702 


.9345 


.9706 


.9352 


.9709 


.9359 


.9712 


.9366 


.9716 


151 


.9373 


.9719 


.9380 


.9722 


.9387 


.9725 


.9394 


.9729 


.9401 


.9732 


.9408 


.9735 


152 


.9415 


.9738 


.9422 


.9741 


.9428 


.9744 


.9435 


.9747 


.9442 


.9751 


.9448 


.9754 


153 


.9455 


.9757 


.9462 


.9760 


.9468 


.9763 


.9475 


.9766 


.9481 


.9769 


.9488 


.9772 


154 


.9494 


.9774 


.9500 


.9777 


.9507 


.9780 


.9513 


.9783 


.9519 


.9786 


.9525 


.9789 


155 


.9532 


.9792 


.9538 


.9794 


.9544 


.9797 


.9550 


.9800 


.9556 


.9803 


.9562 


.9805 


156 


.9568 


.9808 


.9574 


.9811 


.9579 


.9813 


.9585 


.9816 


.9591 


.9819 


.9597 


.9821 


157 


.9603 


.9824 


.9608 


.9826 


.9614 


.9829 


.9619 


.9831 


.9625 


.9834 


.9630 


.9836 


158 


.9636 


.9839 


.9641 


.9841 


.9647 


.9844 


.9652 


.9846 


.9657 


.9849 


.9663 


9851 


159 


.9668 


.9853 


.9673 


.9856 


.9378 


.9858 


.9683 


.9860 


.9688 


.9863 


.9693 


.9865 


160 


.9698 


.9867 


.9703 


.9869 


.9708 


.9871 


.9713 


.9874 


.9718 


.9876 


.9723 


9878 


161 


.9728 


.9880 


.9732 


.9882 


.9737 


.9884 


.9742 


.9886 


.9746 


.9888 


.9751 


9890 


162 


.9755 


.9892 


.9760 


.9894 


.9764 


.9896 


.9769 


.9898 


.9773 


.9900 


.9777 


9902 


163 


.9782 


.9904 


.9786 


.9906 


.9790 


.9908 


.9794 


.9910 


.9798 


.9911 


.9802 


9913 


164 


.9806 


.9915 


.9810 


.9917 


.9814 


.9919 


.9818 


.9920 


.9822 


.9922 


.9826 


9923 


165 


.9830 


.9925 


.9833 


.9927 


.9837 


.9929 


.9841 


.9930 


.9844 


.9932 


.9848 


9933 


166 


.9851 


.9935 


.9855 


.9937 


.9858 


.9938 


.9862 


.9940 


.9865 


.9941 


.9869 


9943 


167 


.9872 


.9944 


.9875 


.9945 


.9878 


.9947 


.9881 


.9948 


.9885 


.9950 


.9888 


9951 


168 


.9891 


.9952 


.9894 


.9954 


.9897 


.9955 


.9900 


.9956 


.9903 


.9957 


.9905 


9959 


169 


.9908 


.9960 


.9911 


.9961 


.9914 


.9962 


.9916 


.9963 


.9919 


.9965 


.9921 


9966 


170 


.9924 


.9967 


.9927 


.9968 


.9929 


.9969 


.9931 


.9970 


.9934 


.9971 


.9936 


9972 


171 


.9938 


.9973 


.9941 


.9974 


.9943 


.9975 


.9945 


.9976 


.9947 


.9977 


.9949 


9978 


172 


.9951 


.9979 


.9953 


.9980 


.9955 


.9981 


.9957 


.9981 


.9959 


.9982 


.9961 


9983 


173 


.9963 


.9984 


.9964 


.9984 


.9966 


.9985 


.9968 


.9986 


.9969 


.9987 


.9971 


9987 


174 


.9973 


.9988 


.9974 


.9988 


.9976 


.9989 


.9977 


.9990 


.9978 


.9991 


.9980 


9991 


175 


.9981 


.9992 


.9982 


.9992 


.9983 


.9993 


.9985 


.9993 


.9986 


.9994 


.9987 


9994 


176 


.9988 


.9995 


.9989 


.9995 


.9990 


.9996 


.9991 


.9996 


.9992 


.9996 


.9992 


9997 


177 


.9993 


.9997 


.9994 


.9997 


.9995 


.9998 


.9995 


.9998 


.9996 


.9998 


.9996 


9998 


178 


.9997 


.9999 


.9997 


.9999 


.9998 


.9999 


.9998 


.9999 


.9999 


.9999 


.9999 


9999 


179 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 


.9999 0.0000 


1.0000 


0000 



INDEX 



Abscissa, 6. 

Absolute value, of a directed quan- 
tity, 7. 

Addition, of angles, 9 ; formulas in 
trigonometo', 95. 

Angle, definition of, 7 ; directed, 7 ; 
measurement of, 8 ; addition and 
subtraction of, 9 ; functions of, 2 ; 
of elevation and depression, 16 ; 
of triangle, 48 ; in artillerj' sen-ice, 
76. 

Annuities, 70. 

Arc of a circle, 76. 

Artillery service, use of angles in, 76. 

Axes, of coordinates, 5. 

Briggian logarithms, 54. 

Characteristic of a logarithm, 54. 
Cologarithms, 59. 
Common logarithms, 54. 
Compass, Mariner's, 29. 
Computation, numerical, 18, 24 ; 

logarithmic, 61 ff. 
Coordinates in a plane, 5. 
Cosecant, 32. 
Cosine, definition of, 12 ; variation 

of, 81 ; graph of, 82 ; law of — s, 

40. 
Cotangent, definition of, 32. 
Course, 29. 
Coversed sine, 32. 

Dead reckoning, 30. 

Departure, 29. 

Difference in latitude, 29 ; in longi- 
tude, 30. 

Directed, angles, 7 ; quantities, 6 ; 
segments, 7. 

Distance, 29. 



Elements of a triangle, 1. 



Function, definition of, 3 ; representa- 
tion of, 32 ; trigonometric, 12 ff., 
58. 

Graph of trigonometric functions, 

80, 82, 83. 

Haversine, definition of, 32 ; solu- 
tion of triangles by, 48 ; tables of, 
117-9. 

Identities, trigonometric, 35. 
Initial position, 7. 
Interest, 70. 
Interpolation, 22. 

Knot, 29. 

Latitude, difference in, 29 ; middle, 
30. 

Law, of sines, 40 ; cosines, 40 ; of 
tangents, 47. 

Logarithm, definition of, 52 ; inven- 
tion of, 50 ; laws of, 53 ; systems 
of, 54 ; characteristic and man- 
tissa of, 54 ; use of tables of, 56 ; 
tables of, 110-16. 

Logarithmic scale, 73. 

Magnitude, 6. 
Mantissa. 54. 
Mariner's compass, 29. 
Middle latitude, 30. 
Mil, 76. 

Napier, J., 50. 
Nautical mile, 29. 
Navigation, 28 fF. 

Negative angle, definition of, 7 ; 
functions of, 85. 



Ordinate, 6. 
121 



122 



INDEX 



Parts of a triangle, 1. 

Period of trigonometric functions, 

80, 82, 84. 
Plane sailing, 28. 
Plane trigonometry, 1. 
Product formulas, 101. 
Projectile, 72. 
Projection, 92. 

Quadrant, 6. 



Radian, 75. 

Radius of inscribed circle. 
Rotation, angles of, 8. 
Rounded numbers, 25. 



46. 



Scale, logarithmic, 73. 

Secant, definition of, 32. 

Significant figures, 25. 

Sine, definition of, 12 ; variation of, 

79 ; graph of, 80 ; law of s, 40. 

Slide rule, 74. 



Solution of triangles, 1, 16 ff., 41 ff., 

48, 62 ff . 
Spherical trigonometry, 1. 

Tables, of squares, 27, 106-7; of 
haversines, 117-9; of logarithms, 
110-11 ; of trigonometric func- 
tions, 112-19. 

Tangent, definition of, 3, 12 ; variation 
of, 82 ; graph of, 83 ; line repre- 
sentation of, 83 ; law of s, 47. 

Triangle, area of, 45 ; angles of, 48 ; 
solution of, 1, 16 ff., 41 ff., 48, 62. 

Trigonometric equations, 88. 

Trigonometric functions, definitions 
of, 3, 12, 15, 32 ; graphs of, 80, 82, 
83 ; computation of, 18 ff. ; periods 
of, 80, 82, 84; inverse, 87; formulas, 
15, 32, 34, 96 ff. ; logarithms of, 
61; tables of, 21, 112-19. 

Versed sine, defined, 32. 



Printed in the United States of America. 



THE following pages contain advertisements of a 
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ELEMENTARY MATHEMATICAL 
ANALYSIS 

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And frank MILLET MORGAN 

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Edited by Earle Raymond Hedrick, Professor of Mathematics 
in the University of Missouri 

///., Cloth, i2mo, $2.60 

A textbook for the freshman year in colleges, universities, and 
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The various subjects are unified by the great centralizing 
theme of functionality so that each subject, without losing its 
fundamental character, is shown clearly in its relationship to the 
others, and to mathematics as a whole. 

More emphasis is placed on insight and understanding of 
fundamental conceptions and modes of thought ; less emphasis 
on algebraic technique and facility of manipulation. Due recog- 
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The text presupposes only the usual entrance requirements in 
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Trigonometry 



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Edited by Earle Raymond Hedrick 

With Brief Tables, 8vo, $1.20 
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The book contains a minimum of purely theoretical matter. Its entire organization is 
intended to give a clear view of the meaning and the immediate usefulness of Trigonometry. 
The proofs, however, are in a form that will not require essential revision in the courses that 
follow. ... 

The number of exercises is very large, and the traditional monotony is broken by illus- 
trations from a variety of topics. Here, as well as in the text, the attempt is often made to 
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The text proper is short; what is there gained in space is used to make the tables very 
complete and usable. Attention is called particularly to the complete and handily arranged 
table of squares, square roots, cubes, etc. ; by its use the Pythagorean theorem and the Cosine 
Law become practicable for actual computation. The use of the slide rule and of four-place 
tables is encouraged for problems that do not demand extreme accuracy. 

Analytic Geometry and Principles of Algebra 

By ALEXANDER ZIWET 

Professor of Mathematics, the University of Michigan 

and 

LOUIS ALLEN HOPKINS 

Instructor in Mathematics, the University of Michigan 

Edited by Earle Raymond Hedrick 

Cloth, i2mo, $1.75 

This work combines with analytic geometry a number of topics traditionally treated in 
college algebra that depend upon or are closely associated with geometric sensation. Through 
this combination it becomes possible to show the student more directly the meaning and the 
usefulness of these subjects. 

The treatment of solid analytic geometry follows the more usual lines. But, in view of the 
application to mechanics, the idea of the vector is given some prominence; and the represen- 
tation of a function of two variables by contour lines as well as by a surface in space is ex- 
plained and illustrated by practical examples. 

The exercises have been selected with great care in order not only to furnish sufficient 
material for practice in algebraic work but also to stimulate independent thinking and to 
point out the applications of the theory to concrete problems. The number of exercises is 
sufl&cient to allow the instructor to make a choice. 



THE MACMILLAN COMPANY 

Publishers 64-66 Fifth Avenue New Tork 



A Short Course in Mathematics 

By R. E. MORITZ 

Professor of Mathematics, University of Washington 

Cloth J 1 2 mo 

A text containing the material essential for a short course in Freshman Mathematics which 
is complete in itself, and which contains no more material than the average Freshman can 
assimilate. The book will constitute an adequate preparation, for further study, and will 
enable the student to take up the usual course in analytical geometry without any handicap. 

Among the subjects treated are : Factoring, Radicals, Fractional and Negative Exponents, 
Imaginary Quantities, Linear and Quadratic Equations ; Coordinates, Simple and Straight 
Line Graphs, Curve Plotting, Maxima and Minima, Areas; The General Angle and Its 
Measures, The Trigonometric or Circular Functions, Functions of an Acute Angle; Solution 
of Right and Oblique Triangles; Exponents and Logarithms; Application of Logarithms to 
Numerical Exercises, to Mensuration of Plane Figures, and to Mensuration of Solids; The 
Four Cases of Oblique Triangles, Miscellaneous Problems Involving Triangles. 

Plane and Spherical Trigonometry 

By LEONARD M. PASSANO 

Associate Professor of Mathematics in the Massachusetts Institute of 

Technology 

Cloth, 8vo, $1.2 j 

The chief aims of this text are brevity, clarity, and simplicity. The author presents the 
whole field of Trigonometry in such a way as to make it interesting to students approaching 
some maturity, and so as to connect the subject with the mathematics the student has pre- 
viously studied and with that which may follow. 

CONTENTS 



CHAPTER 



PL.\NE TRIGONOMETRY chapter 

6. The Solution of General Triangles 



. . 7. The Solution of Trigonometric Equa- 

The Trigonometric Functions of Any tions 



Angle and Identical Relations among 
Them 

2. Identical Relations Among the Func- SPHERICAL TRIGONOMETRY 

tions of Related Angles. The Values 

of the Functions of Certain Angles 8. Fundamental Relations 

3. The Solution of Right Triangles. g. The Solution of Right Spherical Tri 

Logarithms and Computation by angles 

Means of Logarithms lo. The Solution of Oblique Spherical 

4. Fundamental Identities Triangles 

5. The Circular or Radian Measure of an n. The Earth as a Sphere ... 

Angle. Inverse Trigonometric Func- Answers 

tions 



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Publishers 64-66 Fifth Avenue Now York 



Differential and Integral Calculus 

By CLYDE E. LOVE, Ph.D. 

Assistant Professor of Mathematics in the University of Michigan 

Crown 8vo,$2.io 

Presents a first course in the calculus — substantially as the author has 
taught it at the University of Michigan for a number of years. The follow- 
ing points may be mentioned as more or less prominent features of the book : 

In the treatment of each topic the author has presented his material in 
such a way that he focuses the student's attention upon the fundamental 
principle involved, insuring his clear understanding of that, and preventing 
him from being confused by the discussion of a multitude of details. His 
constant aim has been to prevent the work from degenerating into mere 
mechanical routine; thus, w^herever possible, except in the purely formal 
parts of the course, he has avoided the summarizing of the theory into 
rules or formulae which can be applied blindly. 

The Calculus 

By ELLERY WILLIAMS DAVIS 

Professor of Mathematics, the University of Nebraska 

Assisted by William Charles Brenke, Associate Professor of Mathe- 
matics, the University of Nebraska 

Edited by Earle Raymond Hedrick 

Cloth, semi-flexible, with Tables, i2mo, $2.10 
Edition De Luxe, flexible leather binding, $2.50 

This book presents as many and as varied applications of the Calculus 
as it is possible to do without venturing into technical fields whose subject 
matter is itself unknown and incomprehensible to the student, and without 
abandoning an orderly presentation of fundamental principles. 

The same general tendency has led to the treatment of topics with a view 
toward bringing out their essential usefulness. Rigorous forms of demon- 
stration are not insisted upon, especially where the precisely rigorous proofs 
would be beyond the present grasp of the student. Rather the stress is laid 
upon the student's certain comprehension of that which is done, and his con- 
viction that the results obtained are both reasonable and useful. At the 
same time, an effort has been made to avoid those grosser errors and actual 
misstatements of fact which have often offended the teacher in texts other- 
wise attractive and teachable. 



THE MACMILLAN COMPANY 

Publishers 64-66 Fifth Avenue New Tork 



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