Elements Of Plane Trigonometry

TEXT FLY AND

PAGES MISSING WITHIN THE BOOK ONLY

Qi LU

< OU158106>m

=

CD

Osmanfa Call No.

^t^^ M $^

Author Title

University Library "

Accesion

NoJ^-

,

^JIL f

yi *^ ^vU^

j*vucfci+*

6^ %AsV^^q-^A. (NEBRASKA), PH.N.D. (STRASSBURG), PROFESSOR OF MATHEMATICS.

UNIVERSITY OF WASHINGTON

NEW YORK & SONS, WILEY JOHN LONDON-

CHAPMAN & HALL,

INC.

LIMITED

COPYRIGHT, 1910 BY

ROBERT

Printed in

.

.

MORITZ

U.S. A.

Stmbope Jprets G1LSON COMPANY BOSTON. U.S.A.

H.

PREFACE TRIGONOMETRY

is

college

mathematics par

excellence.

To

at least

students college mathematics means trigoIt is important, therefore, that the nometry and nothing else. science be presented in as simple and attractive a manner as possible

90 per cent of

all liberal arts

be made more than a mere method of solving triangles. by making the treatment technical than is customary, by introducing considerable

and that

it

The

the author tries to accomplish

less

first

by not presupposing a too ready knowledge of elementary mathematics, and none at all of the topics ordinarily To accomplish the second point the treated in college algebra. the of the science. central is made idea This permits the angle enrichment of the science through the introduction of a variety of historical matter,

concepts and processes ordinarily reserved for advanced courses in mathematics. Since the treatment departs considerably from that current in textbooks on trigonometry, it is fitting that some of the leading characteristics of the present book should be enumerated at the outset: First, as to subject-matter:

The book has been planned to cover five months' work at four Each month's work is followed by a set of review exercises. Where less time must be given to the subject, (1)

lessons per week.

advanced chapters may, of course, be omitted. the graphic method of splving triangles is intended to impress the need of a more accurate method, certain (2)

the

The introductory chapter on method

of trigonometry. For (3) knowledge of logarithms has not been presupposed. this reason a chapter on logarithms and the use of tables has been

A

incorporated at in logarithms

proper place. Classes who are properly prepared of course omit this chapter.

Many of the more important results have been derived by two more independent methods. This has been done,--

(4)

or

its

may

PREFACE

IV

To give the teacher a choice of To offer the ambitious student

(a)

methods.

the advantage which comes from approaching the same truth from two or more directions. To offer an alternative to the student without a teacher

(b)

(c)

who (5) It is

any one

finds

undue

not intended that

class.

with any one given proof.

difficulty all

The problems

the problems should be assigned to

in each set are carefully graded

and

arranged as follows, (a)

The

(b)

The next

first half in each set are very simple applications of the principles and theorems discussed in the preceding

sections.

three or four problems require some originality on the part of the student. The last few problems in each set are for the more ambitious student and frequently give him the opportunity to dis-

(c)

cover for himself results which are discussed in detail in later sections of the

book.

has been bestowed on the applied problems illusthe solution of right and oblique triangles. In each case trating there is given first a set of problems involving miscellaneous heights (6) Special care

and distances. This is followed by separate sets of applied problems from each of the following sciences: Physics, Engineering, NavigaThese tion, Astronomy and Geography, and Elementary Geometry. arc probably the most varied and complete that have been published in America in recent years.

lists

(7)

than

Trigonometric curves have is

usual.

veloped from

The method

received

much

fuller

of representing functions

first principles.

The treatment

by

treatment

curves

is

de-

includes sine curves

of given amplitude and wave length, logarithmic and exponential curves, composition of harmonic curves, the catenary, and the curve of damped vibrations. (8)

A

special section is

devoted to the angle and

its

functions con-

sidered as functions of time. (9) In developing the theorems of Demoivre and Euler no knowledge of imaginaries is presupposed. The chapter on trigonometric series presupposes no knowledge of series. (10) The usual inadequate treatment of hyperbolic functions is

replaced

by a separate chapter

in

which the analogies between the

PREFACE

V

circular and hyperbolic functions are developed both analytically and geometrically. This chapter concludes with the determination

of the area of a hyperbolic sector. (n) Abundant historical matter has

been introduced throughout

the work.

Second, as to method and arrangement: (12) Logical order has been made subsidiary to order of teaching. To illustrate: the study of the functions of an angle is divided into three parts, (a)

The study

(b)

the solution of right triangles. The study of the functions of the obtuse angle, followed the solution of oblique triangles.

by

(c)

The study

by

(13)

of the functions of

an acute angle, followed by

of the functions of the general angle, followed the solution of trigonometric equations.

The

discussion of the general angle, of circular measure, and two or more angles, is postponed until after the solu-

of functions of

tion of the oblique triangle.

This makes

it

possible to complete the

subject through the solution of triangles in half a semester, an important consideration for classes in short summer sessions, and for engi-

neering students who begin surveying and trigonometry the same Furthermore it removes the suspicion, so often felt by the semester. that the solution of triangles is the sole aim of the science. student, (14) Following the plan long since established in Germany and France, only the three principal functions, sine, cosine and tangent, have received detailed treatment. The corresponding results for

the reciprocal functions are left as exercises for the student.

This

plan economizes time and space and leaves the student with a clearer understanding of the entire subject. (15) Every example worked out in the text is followed by one or more checks. Checks are looked upon as an essential part of every solution. In order to cultivate the use of checks, it would seem best to omit the answers to the exercises, yet every teacher knows the importance of "answers" to the beginner in guiding his first un-

certain efforts. In the present text the plan has been adopted of supplying the answers to a part of the exercises only, with explicit directions to the student to check every problem to which no answer is

given.

This leaves the enforcement of the checking habit largely

PREFACE

VI

with the individual teacher, who may assign as without answers as he deems desirable. (16)

No

many problems

pains has been spared to impress the student with the

limitations in the degree of accuracy, in the answers to problems, imposed by the data, as well as with the limitations in the degree of

accuracy due to the use of tables. Superfluous figures and show of accuracy not warranted by the data or the process of computation employed, are nowhere tolerated in the present book. ^17) On the other hand the student is guarded against disregarding figures and remainders without first measuring the effect of the parts neglected on the required results. Thus, in the computation of

logarithms by means of the logarithmic series, or of natural functions by means of the sine and cosine series, the effect of the neglected part of each series on the final result has been carefully considered in each case.

The book embodies

the author's practical experience of seventeen and univer-

years in presenting the subject to beginners in colleges

His experience has convinced him that the subject of trigonometry can be so simplified and enriched that it deserves the fore-

sities.

most place

year of any high school or the first year of any not college curriculum, only because of its intensely practical value, but chiefly because of its unrivaled cultural value. In the mastery in the last

of logarithms, which strips the most complicated and laborious calculations of their difficulties and irksomeness, the student cannot help

becoming conscious

when properly

of the

tremendous power of the human mind

In the application of algebraic processes and symbols to geometrical and physical magnitudes, he is initiated into a cal

directed.

most far-reaching method

geometry.

The study

of

modern

working knowledge of an indispensable tool activity.

The study

research, that of analyti-

of trigonometric curves should give

of the trigonometric

him a

in every field of scientific

and logarithmic

series,

computation of logarithmic and natural functions, opens an entirely new field of thought with its importance for pracThe actual use of tables familiarizes the student with tical ends.

and

their use in the

the principle of interpolation, a knowledge of which

is

demanded

wherever tables are used. Besides these concepts and processes, the importancb of which all, there are an abundance of others which open the

must appeal to

door to higher realms of thought.

The

simplest applications of

PREFACE

VII

trigonometry to imaginary and complex numbers reveals a new conception of addition and multiplication; in the determination of the roots of unity an otherwise unsolvable problem is solved in all generality; imaginary angles lead to the unsuspected region of hyperbolic functions and reveal a new world of symmetry and its

beauty.

Above

all,

while contemplating the lifelong self-sacrificing efforts of who brought the science to its present state of per-

the master minds

fection, of men who spent their lives without either pecuniary compensation or popular applause in order to share in the building of the temple of abstract truth, the student must come to a better

appreciation of truth for its own sake and be helped in part to a realization of the higher objects of human endeavor.

While writing this book the author has received valuable suggestions from several of his colleagues. Special mention is due Mr. George I. Gavett who supplied some of the applied problems from engineering, script

and

and to Mr. Allen Carpenter who read the

verified

many

entire

of the answers to the problems.

GREEK ALPHABET a pronounced "

y 5 e

" " " "

77

r

"

alpha.

v

beta.

o

gamma. p

epsHon.

v

eta.

T

" " 1

\

" ti

rho.

slgma. "

tan.

" i

upsilon. xi

" ^.V7.

\J/

" 4C

"

beta.

iota.

omikron.

"

' '

I

K

"

PI.

delta.

seta.

pronounced nu.

"

kappa. lambda,

mu.

>>//f.

<j>

X

w

" r:A/.

" owtfga

manu-

CONTENTS CHAPTER

I

INTRODUCTION PACK

ART. 1

.

2.

3.

4.

Graphic solution of triangles Solution of practical problems by the graphic method Inadequacy of the graphic method Definition of trigonometry

CHAPTER

i

4 7 7

II

TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE '

5.

Definition of function

Definition of reciprocal 7. The six trigonometric functions of an acute angle 8. Trigonometric functions determined approximately

9 9 a

6.

9.

10.

Given one of

its

by measurement ....

functions, to construct the angle

Functions of complementary angles

n. Functions

of

14 17

19

o, 30, 45, 60, 90

21

12.

Fundamental

13.

To

25

14.

express each of the functions in terms of a given one Reduction of trigonometric expressions to their simplest form

15.

Trigonometric identities

33

relations

24

CHAPTER

27

III

SOLUTION OF RIGHT TRIANGLES BY NATURAL FUNCTIONS 16.

Table of natural functions

18.

To To

19.

Accuracy

17.

35

find the natural functions of find the angle less

an angle

less

than 90

than 90 corresponding to a given natural function

of results

35

39 43

Solution of right triangles by natural functions 21. Triangles having a small angle 20.

45

49

22.

Historical note

50

23.

Review

51

CHAPTER

IV

LOGARITHMS 24.

Definition of logarithm

53

25.

Fundamental laws governing logarithms Logarithms of special values

54 55

26.

iz

CONTENTS

X

PAGE

ART. 27.

The common system

28.

Rule

29.

Table of

30.

$\

.

To To

of logarithms

57

for the characteristic

common

58 61

logarithms

number number corresponding to a given logarithm

find the logarithm of a given

62

find the

65

32.

Directions for the use of logarithms

67

33.

Application of logarithms

72

34.

To compute a

table of

common

logarithms

73

N

36.

Relation between loga and log&N Natural or hyperbolic logarithms

37.

Tables of logarithmic trigonometric functions

35.

38. 39.

To To

75

76 78

find the logarithmic trigonometric functions of

an angle

less

than 90

function

81

4o: Logarithmic functions of angles near o or 90 41. Use of 5 and T table 42.

79

find the angle corresponding to a given logarithmic trigonometric

83

84

Historical note

87

CHAPTER V LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 43.:

Logarithmic solution of right triangles

44.

Number

45.

Applied problems involving right triangles

Heights and distances. 47. Problems for engineers

..

46.-

48.

.

.

.

89

.

of significant figures

93

94

.

95

.

.

96

Applications from physics

Problems

98 100

.

.

in navigation Geographical and astronomical problems 51. Geometrical applications 52. Oblique triangles solved by right triangles

49.

.

.

103 106

50.

CHAPTER

no

VI

FUNCTIONS OF AN OBTUSE ANGLE 53.

Rectangular coordinates

54.

Definitions of the trigonometric functions of

55.

The

56.

Fundamental

signs of the functions of

any angle

less

than 180 ....

an obtuse angle

119

relations

Functions of supplementary angles 58. Functions of (90 -f- 0) SQ. Functions of 1 80.

....

57.

.

.

.

.

.

60.

Angles corresponding to a given function

61.

Review

119 120 121

.

.

117 1 18

... .,

121

.

.

122 123

CONTENTS

CHAPTER

xi

VII

PROPERTIES OF TRIANGLES PAGE

ART.

62. 63. 64.

The law of sines ............................................ The projection theorem^ ............................... The law of cosines .^-A? ...... ........... .. .

...

Arithmetic solution of triangles .. 66. The law of tangents. .$*??. ....... i/ 67. Formulas for the area of a triangle .VT. ...... 65.

.

......

.

68.

127 128

130

.132

.

Functions of half the angles in terms of the sides .............

CHAPTER

125 126

134

VIII

SOLUTION OF OBLIQUE TRIANGLES 69. 70. 71.

72. 73.

74.

Solution of oblique triangles

.

I.

them

138 141

....

.

138

144 147

150

Auxiliary geometrical constructions

.

152

.

79.

80.

Geometrical applications

77.

...

150

System of simultaneous equations Miscellaneous heights and distances. $fi Applications from physics Applications from surveying and engineering. Applications from navigation Problems from astronomy and meteorology

78.

of

.

Practical applications (a) System of triangles

(c)

76.

.

.

(6)

75.

...

.

.

Given two angles and one side Case II. Given two sides and the angle opposite one Case III. Given two sides and the included angle Case IV. Given three sides Case

.

.

.

155

.

.

V

.

158 161

164

.

171 .

.

,

.

.

.

172

174

CHAPTER IX THE GENERAL ANGLE AND

ITS

MEASURES

81.

General definition of an angle

82.

Positive and negative angles

83.

Complement and supplement

178

84.

Angles in the four quadrants

178

85.

Sexagesimal measure of angles

86.

Decimal division

of degrees

..

177

178

.

179 180

87.

Centesimal measure of angles

180

88.

The

181

circular or natural

system of angular measures 89. Comparison of sexagesimal and circular measure 90. Relation between angle, arc and radius 900. Area of circular sector 91.

Review

182

185 187

189

CONTENTS

Xll

CHAPTER X FUNCTIONS OF ANY ANGLE PAGB

ART. 92. Definition of the trigonometric functions of any angle 93. Signs of the functions in each of the quadrants

191

Periodicity of the trigonometric functions 95. Changes in the value of the functions 96. Changes in the value of the tangent

193

192

94.

193 195

of results

97.

Summary

98.

Fundamental

195

relations

196

Representation of trigonometric functions by lines TOO. Reduction of the functions to the first quadrant

196

99.

101. 102.

Reductions from the third quadrant Reductions from the fourth quadrant

199 201

104.

Functions of negative angles Table of principal reduction formulas and general rules

105.

Generalization of the preceding reduction formulas

103.

199

203

204 206

CHAPTER XI FUNCTIONS OF

Two

OR MORE ANGLES

106.

Addition theorem for the sine and cosine

107.

Generalization of the addition theorems

108.

Addition theorems.

109.

Subtraction theorems

209 210 211

Second proof f< -r

the sine and cosine

no. Tangent of the .um and difference of two angles in. Functions of double an angle 112. Functions of half an angle 113. Sums and differences of sines or cosines transformed into products.

CHAPTER

212 215 216

216 ...

219

XII

TRIGONOMETRIC EQUATIONS 114.

Angles corresponding to a given function

225

115.

Principal value

225 226

1

16.

1 1 7.

1 18.

Formula for angles having a given sine Formula for angles having a given cosine Formulas for angles having a given tangent of results

i

IQ.

Summary

1

20.

Trigonometric equations involving a single angle

226 227 227 227

121.

Trigonometric equations involving multiple angles

233

122.

Trigonometric equations involving two or more variables

236

123.

Solutions adapted to logarithmic computation Inverse functions

240

124.

125.

Review

250

2 45

CONTENTS

CHAPTER

Xlil

XIII

TRIGONOMETRIC CURVES PAGE

ART. 126.

127. 1

28.

129. 130. 131. 132.

Functions represented by curves

253

The straight line The circle The hyperbola The sine curve The tangent curve The sinusoidal or simple harmonic curves

253

254 255 255 257

260

Angles as functions of time 134. Composition of sinusoidal curves

262

133.

135.

Theorem on composition

264

of sinusoidal

curves having equal

wave

lengths 136.

137. 138. 139.

140. 141. 142.

267 268

Fourier's theorem

The logarithmic curve The exponential curve The general exponential curve The compound interest law The catenary The curve of damped vibrations

26q

270 270 272

274 275

CHAPTER XIV TRIGONOMETRIC REPRESENTATION OF COMPLEX NUMBERS Imaginary numbers Geometric representation of imaginary numbers Geometric representation of complex numbers Trigonometric representation of complex numbers Geometric addition and subtraction of complex numbers Physical applications of complex numbers

278

149.

Historical note

285

^50.

Multiplication and division of complex numbers

151.

Powers of complex numbers Roots of complex numbers

287 288

143. 44.

f45.

146. 147. 148.

152.

153. f 54.

To To

n solve the equation z n solve the equation z

i

=

The cube The cube

292

o

293

157. 158.

The

1

56.

159. [60.

To To

295

number

irreducible case

express sin n$ and cos express cos 6 and sin

angles

282 283

o

roots of unity roots of any real or complex Solution of cubic equations

f 55.

281

289 i

-f-

278 280

295

296

298 iiO

in in

powers of sin 6 and cos terms of sines and cosines of multiple

301

302

CONTENTS

XIV

CHAPTER XV TRIGONOMETRIC SERIES AND CALCULATION OF TABLES ART.

PACE

161.

Definition of infinite series

162.

Convergent and non- convergent

163.

Absolutely convergent series

164.

Sum

165. 1 66.

The limit or r n as n approaches The geometric infinite series

167. 168.

169. 170.

171.

of

an

306 307 308 310 310 310

series

infinite series

Convergency test ... Convergency of special

infinity

31 ! series

313

The number e The exponential series The logarithmic series

316

Calculations of logarithms Errors resulting from the use of logarithms .. sin x tan .v . T f as x approaches zero 174. Limiting values of the ratios

172. 173.

.

.

,

.

,

>

x

175.

Limiting values of cos

w -

/ sm

and

n

\

,

x \ x / n )\

x/n

as H approaches infinity

The

177.

Computation of natural functions table Approximate equality of sine, tangent and radian measure of small

178.

and tangent

326 328

/

176.

sine, cosine

318 320 321 324

series

angles

329 330 333

CHAPTER XVI HYPERBOLIC FUNCTIONS 170. 180. 181.

Series with

complex terms

336

Definition of the trigonometric functions of complex Euler's theorem, ei0 = cos 6 i sin

+

Geometrical representation of Euler's theorem 183. Exponential form of the sine and cosine

182.

Hyperbolic functions defined Duality of circular and hyperbolic functions 186. Table of formulas 187. Inverse hyperbolic functions

184.

.

.

.

185.

1 88.

189. 190. 191.

Geometrical representation of hyperbolic functions Area of hyperbolic sector

Use of hyperbolic functions Review

numbers

337 338

339 340 342 343 345 348 349 350 353 354

PLANE TRIGONOMETRY CHAPTER

I

INTRODUCTION In order to work the exercises in this chapter the student should be provided with a pair of compasses, a protractor, and a graduated ruler divided into tenths of a unit.

In plane geometry it is shown 1. Graphic Solution of Triangles. that the six parts (three sides and three angles) of any plane triangle are so related that any three parts suffice to determine the shape of the triangle, and if one of the known parts is a side, the size of the

Furthermore it is shown how to contriangle is also determined. All struct the triangle when a sufficient number of parts is given. four cases. of or the one another under come cases following possible

To

construct the triangle

One

when

there

is

given,

and two angles. and an angle opposite one III. sides and the included angle. IV. Three sides. I.

Two Two

II.

Usually measures. the actual

side

sides

ment

A

them.

we have given not the actual lines and angles but their From these measures lines and angles corresponding to lines and angles may then be constructed by means of

suitable instruments. 1.

of

Such instruments

are,

graduated straight-edge for the construction and measure-

of straight lines of definite lengths.

The

smallest divisions of

the straight-edge should be decimal, either millimeters or tenths of

an inch.

A

2. pair of compasses for the construction of circles cular arcs.

3.

A

protractor for the construction

angles of definite magnitudes.

and measurement

and

cir-

of plane

PLANE TRIGONOMETRY

EXAMPLE

i.

It

is

[CHAP,

i

required to construct a triangle which has two and 1.75 inches respectively and the included

sides equal to 2.5 inches

angle equal to 36. Solution. By means of the protractor construct an angle (Fig. i) equal On to 36. measure off equal

MAN

AM

to

2.5

inches.

AB

On

AN

measure

off

AC

equal to 1.75 inches. Join B and C by a straight line, ABC is the required Fig.

i.

triangle.

The numerical values of the parts which were not known at the outset may now be found by measurement. BC is thus found to be 1.49 inches, and by means of the protractor, angles B and found to be approximately 43.5 and 100.5 respectively. If it is

not possible or convenient to construct the triangle

C

are

full size,

a similar triangle may be constructed on a reduced scale; that is, any unit or a fraction of a unit on the scale may be taken to represent

Thus lines 3 and 4 inches long in the problem. of a be in solution the triangle whose sides are 30 and may employed 40 miles respectively. The angles of the reduced triangle will of any unit occurring

course be equal to the angles of the triangle represented. Similarly, the unknown parts of a triangle which is too small for actual construction, say some microscopic triangle, may be found by measurement from a similar triangle drawn on an enlarged scale.

EXAMPLE

2.

One

side of a triangle measures

adjacent to this

side

measure 23

angles Find the remaining parts of the triangle. Let J inch represent 100 Solution. drawn 3 inches miles. Then a line

600 miles, and the

and 100

respectively.

AB

long will represent 600 miles. At B draw the angles BAC and

and 100

AC

A and

ABD 23 BD intersect

Let respectively. at E. will represent the re-

ABE

Fig. 2.

quired triangle.

Angle

E

measures 57, which of course could have been found sum of the angles .4 and B from 180.

otherwise by subtracting the

AE

and

BE

are found

inches respectively.

to

measure approximately 3.52 and 1.40

Remembering that each J inch

represents 100

INTRODUCTION

i]

miles, the actual lengths represented

by

3

AE

BE

and

are approxi-

mately 704 and 280 miles

respectively. Solutions, like the foregoing, in which geometrical drawings to a scale are employed instead of numerical calculations, are called

graphic solutions.

EXERCISE

i

1. Review the following propositions in geometry. A, B, C represent the three angles of any triangle and a, 6, c the sides opposite these angles.

c.

Given A, B, c\ to construct the triangle. Given a, &, C; to construct the triangle. Given a, b, c; to construct the triangle.

d.

Given

e.

Under what conditions

a. b.

a, b,

A

To

solutions?

The 2.

to construct the triangle.

;

Given a

=

5,

=

4, c

=

7; find

give rise to two different

Given 6

=

4, c

=

C = 90;

5,

Given

b

=

=

600,

the graphic method.*

B=

30',

34,

find the third side

Ans. a

270, c

by

the angles to the nearest

A = 44

to the nearest 15'. 4.

(d)

following problems are to be solved

Ans. 3.

will

only one solution?

A = 100;

=

3,

5.

=

0.029,

Given a

=

42, b

32

=

51,

136 Ans.

Ans.

of the remaining parts.

c

7.

>1

=

44

30',

B=

57,

C=

= =

700.

45'; find the remaining b 0.081, c 0.104.

find the

or c

Given

and the angles

find the third side correct

15', C=

A = 55;

15'.

30'.

Ans. a

B=

sides. 6.

101

A = 37, B = 53.

to the nearest integer.

Given a

C=

approximate measures

33.6, 24.9,

B= B=

84, 96,

C = 41; C = 29.

78 30'; find the ratios between

the sides opposite these angles.

Ans. Approximately a

:

b

:

c

=

5

:

6

:

7.

*

In order to employ the graphic method successfully the student must pracTwo pencils of medium hardness should be used, one sharpened to a point for marking distances, the other sharpened like a chisel for drawing The pencil points are easily kept sharp with the aid of a piece of fine sandlines. tice

accuracy.

The lines should be drawn sharply and they should bisect the points Through which they are intended to pass. In measuring the required parts, beginners should estimate angles to quarters of a degree and lengths to quarters of

paper.

the smallest division of the scale.

PLANE TRIGONOMETRY

[CHAP,

i

Solution of Practical Problems by the Graphic Method. important practical problems, in which a high degree of accuracy is not essential, can be easily solved by the graphic method. 2.

Many

AB

across a Suppose it is required to find the approximate distance lake or'swamp, without actually measuring it. This may be accomplished in various ways, one of which is as follows: Select

A and B tances

some point

P

from which both

are visible, and measure the dis-

AP

and

BP

and

also

the

angle

APB*

This gives two sides and the included angle of the triangle APB from

which

AB may be

found by the method of

the preceding article. Similarly the heights of towers and trees Fig. 3-

and mountains, of clouds and shooting distances the stars, through impenetrable forests across swamps and through mountains, the widths of rivers, ravines and canyons, may be

Even the

determined.

distances between celestial objects may be method after certain other distances

the graphic

approximated by and angles have been measured.

EXAMPLE

In order to determine the width of a

i.

river,

the dis-

tance between two points A and B close to the bank of the river was measured and was found to be 600 feet. The angles BAP and

ABP, formed

with a point

P

Select a suitable scale, say

Solution.

bank and 36

close to the opposite

were also measured and were found to be 50 Required the approximate width of river. i

of the river,

respectively.

inch

to 100 feet, and construct a triangle ABP, = 6 inches and the adjacent having to 50 and 36 respectively. From angles equal

AB

P

PT

perpendicular to AB. the width of the river. represent

draw

PT.

PT

will

represents 100

be found to measure feet,

DEFINITIONS. observer.

will

2.7 inches,

the width of the river

Let

Through

PT

P

Fig. 4-

Measure is

and 270

since each inch feet.

P

the position of the be any point and and draw a vertical line, draw through

a horizontal line meeting the vertical line in H. * The angle between two visible objects is readily measured by means instrument called a transit.

of

an

INTRODUCTION

P is above H, as in the upper figure, the angle HOP is called the angle of elevation of the point P as seen from O, If P is below H, as in the lower figure, the angle #OP is called the angle of depression of the point P as seen from O. If

It is obvious that the angle

of elevation or

depression of an object depends upon the position of the observer.

of

EXAMPLE

2.

From a

point

P at

M

FiS-

5-

the foot of a mountain, the angle

measured and is found to be 30; after walking two miles toward the summit on an incline averaging 15, the angle of elevation is found to measure 45. Required the

summit

of elevation of the

is

height of the mountain. Solution. Draw a horizontal

PX.

line

angle

XPN =

Construct an

30; then

PN

represents the direction in which the summit of the

mountain

is

from P.

seen

Construct angle XPC = 15, and take PC two units in

. _. Fig. 6.

length. Then, if each unit the one will C mile, position from which the represent represents second observation was made. Through C draw CX' parallel to PX, and construct an angle X'CN'

=

45.

Then CN'

represents the direction in which the

summit

is

seen from C. Since the

summit

is

on each

of the lines

M. PX. Then

PN

and CN',

it

must be

located at their point of intersection

Draw

MF

perpendicular to

MF

will represent

the

mountain on the same scale on which PC represents two miles. Measure MF. If PC was taken equal to 6 inches, MF will measure 5.8 inches. Since 3 inches represents one mile, rep-

height of the

MF

resents 1.933 miles, or 10,200 feet approximately.

EXERCISE

2

The following problems are to be solved graphically. The student is expected to obtain distances correct to three figures and angles correct to nearest

15'-

PLANE TRIGONOMETRY

6

[CHAP,

i

1. At a distance of 400 feet from the foot of a tree, the top of the tree subtends an angle of 20. Find the height of the tree. Ans. 145.6 ft.

A

B

2. 8 miles disstraight road leads from a town A to a town tant; another road leads from A to a third town C 10 miles distant.

The 3.

angle between the roads

What

is

is

How

65.

far is it

from B to C ? Ans. 9.82 mi.

(= angle of elevation) of the sun, when a high casts a shadow 190 feet long on a horizontal

the altitude

building 75 feet

Ans. 21

plane?

30'.

The

great pyramid of Gizeh is 762 feet square at its base and each face makes an angle of 51 51' with the horizontal plane. 4.

Determine the height of the pyramid, assuming that it comes to Ans. 485 feet. an apex. 5. As a matter of fact, the pyramid mentioned in Problem 4 does not come to a point, but terminates in a platform 32 feet square. Find the actual height of the pyramid. Ans. 465 feet. 6.

line

An

observer on board ship sees two headlands in a straight The ship sails northwest for 5 miles, when one of the

N. 35 E.

headlands appears due east and the other due northeast. apart are the headlands? 7.

How

far

Two observers

at the

on opposite sides of a balloon observe the balloon same instant and find its angles of elevation to be 56 and 42

The observers are one mile apart. Find the height of respectively. at the time the observations were taken. the balloon Ans. 0.6 mi. nearly. 8.

AB

In order to determine the distance across a swamp, a distance laid off 100 yards long, and at each extremity of the line AB

was

the angles were measured between the other extremity of the line and each of two stakes P and Q placed at opposite ends of the swamp. At one extremity of the line the angles measured 35 and 85 respec-

end the angles measured 40 and 121 respectively. Find the distance PQ. 9. Find the perimeter of a regular polygon of 7 sides inscribed in Ans. 60.75 fta circle whose radius is 10 feet. = = = 10. The sides of a triangle are a 10, b 12, c 15 respectively. tively, at the other

Find the

radii of the inscribed

the angles of the triangle.

Ans.

and r

of the circumscribed circles

=

3.23,

R =

7.52,

and

INTRODUCTION

3-4]

7

Inadequacy of the Graphic Method. The graphic method of solving triangles, though exceedingly simple and useful, is not suffiFor instance, in the last problem ciently accurate for all purposes. 2 obtained the results of Exercise by the graphic method are: 3.

r

=

3 .2 3

while the

,

#=7-52,

more accurate

A=

=53,

41 30',

results,

C=8 5

3o

/

,

obtained by a method to be de-

scribed later, are: r

=

3.2331,

=52 The

#=

7-5236,

53' 27",

.4

=