Online Library → Edward Olney → Elements of trigonometry, plane and spherical → online text (page 1 of 28)

Font size

Irving Stringham

ncs.

1.)

"

received

ishers to

ional ap-

) unique,

f calling

asses of

Each volume in the senes is so constructed mat 11 may be used

with equal ease by the youngest and least disciplined who should

be pursuing its theme, and by those who in more mature years

and with more ample preparation enter upon the study.

Sheldon & Company's 2'ext-22oofcs.

Hill's Elements of Rhetoric and Composition

By D. J. HILL, A.M., President Lewisburg University, author

of the Science of Rhetoric. Beginning with the selection of a

theme, this book conducts the learner through every process

of composition, including the accumulation of material, its

arrangement, the choice of words, the construction of sentences,

the variation of expression, the use of figures, the formation of

paragraphs, the preparation of manuscript, and the criticism of

the completed composition.

Hill's Science of Rhetoric

An introduction to the Laws of Effective Discourse. By

D. J. HILL, A.M., President of the University at Lewisburg.

12mo, 300 pages.

This is a thoroughly scientific work on Rhetoric for advanced

classes.

Intellectual Philosophy (ELEMENTS OF). 426 pages

By FRANCIS WAYLAND, late President of Brown Univer-

sity.

The Elements of Moral Science

By FRANCIS WAYLAND, D.D., President of Brown Univer-

sity, and Professor of Moral Philosophy. Fiftieth thousand.

12mo, cloth.

Elements of Political Economy

By FRANCIS WAYLAND, D.D., late President of Brown Uni-

versity. 12mo, cloth, 403 pages.

Recast by AARON L. CHAPIN, D.D., President of Beloit

College.

No text-book on the subject has gained such general accept-

ance, and been so extensively and continuously used, as Dr.

Wayland's. Dr. Chapin has had chiefly in mind the wants of

the class-room, as suggested by an experience of many years.

His aim has been to give in full and proportioned, yet clear

and compact statement, the elements of this important branch

of science, in their latest aspects and applications.

OLNEY'S MATHEMATICAL SERIES.

"ElTEMENTS

TBIGOE"OMETKY

PLANE AND SPHERICAL.

BY EDWARD OLNEY,

MATHEMATICS IN THB UHIVKKSITY Off MICHIGAN

NEW YORK:

SHELDON & COMPANY,

No. 8 MURRAY STREET.

1885-

1

Entered according to Act of Congress in the year 1870. by

SHELDON & COMPANY,

In the Office of the Librarian of Congress at Washington.

PROF, OLNEY'S MATHEMATICAL COURSE,

INTRODUCTION TO ALGEBRA - - -

COMPLETE ALGEBRA

KEY TO COMPLETE ALGEBRA - . - -

UNIVERSITY ALGEBRA

KEY TO UNIVERSITY ALGEBRA -

A VOLUME OF TEST EXAMPLES IN ALGEBRA - - -

ELEMENTS OF GEOMETRY AND TRIGONOMETRY

ELEMENTS OF GEOMETRY AND TRIGONOMETRY, University

Edition

ELEMENTS OF GEOMETRY, separate -

ELEMENTS OF TRIGONOMETRY, separate -

GENERAL GEOMETRY AND CALCULUS -

BELLOWS' TRIGONOMETRY -

PROF, OLNEY'S SERIES OF ARITHMETICS.

PRIMARY ARITHMETIC -

ELEMENTS OF ARITHMETIC - - - -

PRACTICAL ARITHMETIC

SCIENCE OF ARITHMETIC - - - - : - -

E. &. Trig.

CONTENTS.

PART IV. TRIGONOMETRY.

CHAPTER I.

PLANE TRIGONOMETRY.

SECTION I.

DEFINITIONS AND FUNDAMENTAL RELATIONS BETWEEN THE TRIGONOMETRI-

CAL FUNCTIONS OP AN ANGLE (OR ARC). PAGE.

Definitions 1-5

Fundamental Relations of the Trigonometrical Functions of an

Angle 5-7

Signs of the Functions 7-10

Limiting Values of the Functions 10-14

Functions of Negative Arcs 15-16

Circular Functions 10

Exercises 17-20

SECTION II.

RELATIONS BETWEEN THE TRIGONOMETRICAL FUNCTIONS OF DIFFERENT AN-

GLES (OR ARCS).

Functions of the Sum or Difference 20-25

Functions of Double and Half Angles 26

Exercises 26-29

SECTION III.

FORMULAE for rendering Calculable by Logarithms the Algebraic Sum

of Functions 29-30

Exercises 30-31

SECTION IV.

CONSTRUCTION AND USE OF TABLES.

Definitions 31-32

To Compute a Table of Natural Functions 32-33

To Compute a Table of Logarithmic Functions 33-34

Exercises in the Use of the Tables 34-39

Functions of Angles near the Limits of the Quadrant 39-42

Exercises . . 43

800561

IV CONTENTS.

SECTION V.

SOLUTION OF PLANE TRIANGLES. PAGE.

Of Right Angled Triangles 44-45

Exercises and Examples 45-48

Practical Applications 48-49

Of Oblique Angled Plane Triangles 49-51

Exercises 51-55

Oblique Triangles Solved by means of Right Angled Triangles . . . 55-56

Exercises 56-57

functions of the Angles in Terms of the Sides 58-59

Exercises 59-00

Area of Plane Triangles 60-61

Practical Applications 61-64

CHAPTER II.

SPHERICAL TRIGONOMETRY.

INTRODUCTION".

PROJECTION OP SPHERICAL TRIANGLES.

Definitions and Fundamental Propositions 65-66

Projection of Right Angled Spherical Triangles 66-71

Projection of Oblique Angled Spherical Triangles 71-73

SECTION I.

SOLUTION OP RIGHT ANGLED SPHERICAL TRIANGLES.

Definitions , 73-75

Exercises 75-76

Napier's Rules 76-78

Determination of Species 79-81

Exercises in Solution of Right Angled Spherical Triangles 81-84

Quadrantal Triangles 84-85

SECTION II.

OP OBLIQUE ANGLED SPHERICAL TRIANGLES.

To find the Segments of a Side made by a Perpendicular let fall

from the opposite angle 85-86

The Relation of the Sides and Opposite Angles 87

Solution of Oblique Spherical Triangles by Napier's Rules, in

Three Problems 87-90

Exercises .... 90-95

CONTENTS. V

SECTION III.

GENERAL FORMULAE. PAGE.

Angles as Functions of Sides, and Sides as Functions of Angles. 96-100

Gauss's Equations 100-101

Napier's Analogies 102

Exercises in the Use of these Formulae 102-107_

SECTION IV.

Aiea of Spherical Triangles 108-110

Practical Applications of Spherical Trigonometry 110-113

TABLES.

Introduction to the Table of Logarithms 1-10

Table of Logarithms of Numbers 11-28

Table of Natural Sines and Cosines, and of Logarithmic Sines, Cosines,

Tangents, and Cotangents 29-74

Table for Precise Calculation of Functions near their Limits 75-78

Table of Tangents and Cotangents 79-88

PART IV.

TRIGONOMETRY,

CHAPTEE L

PLANE TRIGONOMETRY.

SECTION L

DEFINITIONS AND FUNDAMENTAL RELATIONS BETWEEN THE

TRIGONOMETRICAL FUNCTIONS OF AN ANGLE (OR ARC).

1. Trigonometry is a part of Geometry which has for its sub-

ject-matter, Angles. It is chiefly occupied in presenting a scheme

for measuring and comparing angles, by means of certain auxiliary

lines called Trigonometrical Functions^ investigating the relations

between these functions, and in the solution of triangles by means

of the relations between their sides and the trigonometrical functions

of their angles.

2. Plane Trigonometry treats of plane angles and triangles,

in distinction from Spherical Trigonometry, which treats of spherical

angles and triangles.

3. A Function is a quantity, or a mathematical expression,

conceived as depending upon some other quantity or quantities for

its value.

ILL'S. A man's wages for a given time is a function of the amount received

per day ; or, in general, his wages is a function of both the time of service and

the amount received per day. Again, in the expressions y = 2ax*, y = x* -

2bx + 5,y = 2 log ax, y = a x ,y is a function of a: ; since, the numbers 2, 5, a and b

being considered fixed or constant, the value of y depends upon the value we

assign to x. For a like reason such expressions as \/a a x* , and 3oa; 3 2v/a"j

may be spoken of as functions of a:. Once more, the area of a triangle is a func-

tion of its base and altitude.

4. Angles as Functions of Arcs. Wo have learned in

Geometry (PART II., SEC. VI.), that angles and arcs may be treated

as functions of each other; and that, if the angles be taken at the

I

2 PLANE TRIGONOMETRY.

centre of the same or equal circles, the arcs intercepted nave the

game ratio as the angles themselves, and hence may be taken as then

measures or representatives. For trigonometrical purposes, an angle

is considered as measured by an arc struck with a radius 1, from the

angular point as a centre.

5. A Degree being the -g-J-^ part of the circumference of a circle,

becomes the measure of -fa of a right angle; and, for convenience, it

is customary to speak of such an angle as an angle of one degree, of

four times as large an angle as an angle of four degrees, etc., apply-

ing the term directly to the angle. A small circle written at the

right and a little above a number indicates degrees ().

6. A Minute is fa part of a degree. Minutes are designated by

an accent ('). A Second is -fa part of a minute. Seconds are

indicated by a double accent ("). Smaller divisions of angles (or

arcs) are most conveniently represented as decimals of a second

thougn the designations thirds, fourths, etc., are sometimes met with,

and signify further subdivisions into 60ths. 5 12' 16" 13'" is read,

" 5 degrees, 12 minutes, 16 seconds, and 13 thirds."

ILL'S. In Fig. I AOB is an angle of 35, because the measuring arc ab

contains 35 of the 360 equal parts into which

the circumference whose radius is 0a, could be

divided. In like manner BOC is an angle of 7.

BOC = iAOB = iAOC. Hence, it becomes evi-

dent that we may use the numbers 35, 7, and 42

to represent the respective angles AOB BOC

and AOC, or the corresponding arcs ab, be, and

cue.

7. A Quadrant is an arc of 90,

and is the measure of a right angle;

hence, a right angle is called an angle of

**> * 90. Thus arc ad, Fig. 1, = 90. or angle

AOD = 90.

8. The Complement of an angle or arc is what remains aftei

subtracting the angle or arc from 90. The Supplement of an angle

or arc is what remains after subtracting the angle or arc from 180

ILL'S. In Fig. 1, the angle BOD is the complement of AOB, and the arc 3d

is the complement of arc ab. The complement of 35 is 90 35- 55. Thf

supplement of 35 is 180- 35= 145.

DEFINITIONS AND FUNDAMENTAL RELATIONS. 3

,9. A Quadrant is often represented by ^T, since <n is the semi-

cirou inference when the radius is unity. When this notation is used,

180

the Unit Arc becomes - = 57.29578 nearly, or 57 17' 44 ".8 +,

it

which is an arc equal in length to the radius.

10. For trigonometrical purposes, an angle is conceived as ger

erated by the revolution of a line about the angular point, ana

hence may have any value whatever, not only from to 180, but

from to 360, and even to any number of degrees greater than

360, as 1280, etc. An angle of 45 is generated by of a revolu-

tion, 90 by J of a revolution, 180 by i a revolution, 270 by } of a

revolution, 360 by one revolution, 450 by 1J revolutions, 1280 by

3$ revolutions, etc., etc.

11. In accordance with the conception of an angle as generated

by a revolving line, the measuring arc is considered as originating

at the first position of the revolving line (i. e., with one side of the

angle), and terminating in the line after it has generated the angle

under consideration (i. e., with the other side of the angle), The

first extremity is called the Origin of the arc, and the other the

Termination.

ILL'S. In Fig. 1, let the angle AOB be considered as generated by a line

starting from the position OA, and revolving around the point O, from right to

left,* till it reaches the position OB. Oa being taken as unity, the arc cub is the

measuring arc of the angle AOB ; a is its origin, and b its termination.

12. In the generation of angles by means of a revolving line, the

normal motion is considered to be from right to left, and the quad-

rants are numbered 1st, 2d, 3d, and 4th, in the order in which they

are generated.

13. The Trigonometrical Functions are eight in num-

ber; viz., sine, cosine, tangent, cotangent, secant, cosecant, versed-

sine, and cover sed- sine. These lines are functions of angles, or.

what amounts to the same thing, of arcs considered as measures of

angles, and are the characteristic quantities of trigonometry.

14. The Sine of an angle (or arc) is a perpendicular let fall

from the termination of the measuring arc upon the diameter passing

through the origin of the arc. Thus in Fig. 2, bd is in each case

the sine of the angle AOB, or of the arc axb.

* The pupil will understand that, if he imagines himself standing at the centre of moti .n.

a* the moving body or point passes before him, the distinctions " from right to left." and

14 from left to right," are easily made.

PLANE TKIGONOMETHY.

15. The Trigonometrical Tangent of an angle (or arc)

is a tangent drawn to the measuring arc at its origin, and limited

(a)

CL- A

(c)

by the produced diameter passing through the termination of the

arc. Thus in Fig. %, ac is in each case the tangent of the angle AOB,

or of the arc axb.

16. The Secant of an angle (or arc) is the distance from the

angular point, or centre of the measuring circle, to the extremity of

the tangent of the same angle (or arc). Thus in Fig. 2, Oc is in

each case the secant of the angle AOB, or of the arc axb.

17. The Versed-Sine of an angle (or arc) is the distance

from the foot of the sine of the same angle (or arc) to the origin of

the measuring arc. Thus, in Fig. 2, da is in each case the versed-

sine of the angle AOB, or of the arc axb.

18. The prefix co, in the names of the four trigonometrical func-

tions in which it occurs, is an abbreviation for the word complement.

Thus cosine means complement-sine, i. e., the sine of the comple-

ment; cotangent means tangent of the complement; etc. The co-

sine of 40 is the sine of 90 40, or 50 ; the cosine of 110 is the

sine of 90 110, or 20; the cotangent of 30 is the tangent of

60 ; the cosecant of 200 is the secant of - 110.

j.9. Construction of the Complementary Functions.

Let us now see how the complementary functions are constructed with refer-

uce to their primitives, premising that all arcs in ffy. 3, reckoned from A, are

DEFINITIONS AND FUNDAMENTAL RELATIONS.

to foe reckoned around from right to left in this discussion. 1st. Let AP be

uny arc less than 90 ; then 90' AP = aP is its complement. Now considering

a as the origin and P the termination of

this complementary arc, Pd is its sine, at

its tangent, Ot its secant, and ad its versed -

sine. Hence, Pd, at, Ot, and ad are respect-

ively the cosine, cotangent, cosecant, and

coversed-sine of the arc AP, or the angle

AOP. 2d. Letting APP' be any arc between

90 and 180, its complement is 90 APP'

or aP', the sign signifying that the arc

is reckoned backward from P f to a. But as

the values of the functions will be the same

whether the origin be taken at P' or at a,

we may take a as the origin of this comple-

mentary arc, and P' as its termination,

whence P' d' becomes its sine, at' its tan-

gen t, Ot' its secant, and ad' its versed-sine.

Therefore P'd', at', Of, and ad', are respect-

ively the cosine, cotangent, cosecant, and Fl - 3.

coversed-sine of the arc APP', or the angle AOP'. 3d. In like manner, aP' is

shown to be the complement of arc APP'P" ; and as P"d", at, Ot, and ad!"

are respectively the sine, tangent, secant, and versed-sine of this complement,

they are the corresponding cofunctions of the arc APP'P", or the salient angle

AOP" 4th. In the same way, it appears that P r "d'", at', Of, and ad'" are the

cosine, cotangent, cosecant, and coversed-sine of the arc APP'P" P'", or the

salient angle AOP'". Observe that a, a point on the measuring arc 90 from the

primitive origin, is the origin of all the complementary functions.

SCH. It will readily appear from the figure that the cosine of an angle

(or arc) is always equal to the distance from the foot of tJie sine to the vertex of the

angle (or the centre of the measuring arc). This is the more convenient prac-

tical definition. Thus the cosine of AP is Pd = DO; the cosine of APP' is

P'd 1 = D'O, etc.

20. Notation. Letting x represent any angle (or arc), the

several trigonometrical functions of it are writteL sin a:, cos a;, tana;,

cots, sec a;, cosecz, versa;, and covers x. They are read "sinea,"

" cosine x" " tangent x" " cotangent <B," etc.

FUNDAMENTAL RELATIONS BETWEEN THE TRIGONOMETRICAL

FUNCTIONS OF AN ANGLE (OR ARC).

[Note. These fundamental relations rant he made perfectly familiar They must be

memorized, and be as familiar as the Multiplication Table. The student can do nothing in

trigonometry without them.]

Hf The discussions in this treatise all proceed upon one general

plan ; viz., First obtain the particular propwty of the

6

PLANE T1UGONOMET11Y.

sine and cosine, and from this deduce all the other*

according to the dependencies shown in the follow-

ing proposition.

21. Prop. The Fundamental Relations which the Trigono-

metrical Functiw* *ustain to each other are:

(1) sin 2 x + cos 9 x = 1 ;

. . sin a;

(2) tan x= ;

v ' '

/ox cos a;

(3) cot x = -r

sma

1

(4)

tan x '

(5) seca; = :

cos a;

(6) 1

. ' sin a;'

(7) sec 2 x = 1 '4- tan 8 x ;

(8) cosec'a; = 1 + cot 8 a;

(9) vers x = 1 cos x ;

(10) covers # = 1 sin a:.

(The forms sin a a;, sec 2 a;, etc., signify the square of the sine, the

square of the secant of x, etc., and are read " sine square x," " secant

square x" etc. The student should distinguish between sin a a;, and

sin a; 2 .)

DEM. In Fig. 4, let x represent any arc as AP, less than 90. Then PD = sin *,

OD or Pd = cos#, AT = tana?, OT = seca, at = cotz, OZ = cosecz, AD = versin *,

and ad = co versin x.

FIG. 4.

= ; but op

'

(1). In the right-angled triangle POD

PD 3 + OD 2 OP 2 > 01> sin 2 a; + cos 2 a; = 1,

since OP = radius = 1.

(2). From the similar triangles POD and

TOA,

AT _ PD f> _ sin a;

OA OD' ~cosa;.

(8). From the similar triangles POd and

tOa,

at Pd cos x

= 7r-3, Or COtiC -: .

Oa Od sin x

(4). Multiplying (2) and (3) together,

sin a; cos a; 1

tan#cot2:= : = 1, 01 tan;? =

cos x sin x cot*

(5). From the similar triangles OTA and

OPD,

1

' * ' ~~ COS X

DEFINITIONS AND FUNDAMENTAL RELATIONS.

(6). From the similar triangles Ota and OPd,

Of OP 1

TT- = X-TJ or cosec# = - .

Oa Od' sin a-

(7). From the right-angled triangle OAT,

OT a = OA a -f AT 2 , or sec'z = 1 + tan 1 *.

(8). From the right-angled triangle

0^ = 2 + otf 2 , or cosec' 2 ^ = 1 4- cot a z.

(9). AD = AO - OD, or vers x = 1 - cos x.

(10). ad = aO Od, or covers x 1 sin x.

Thus the fundamental relations of the functions are established for an are

less than 90. But it will readily appear that the relations are the same for any

other arc. For example, let x = AP' be any arc between 90 and 180. Then

the triangle P'D'O gives sin a # + cos 2 z 1, since P'D' = sin #, and OD' = cos x.

The similar triangles P'D'O and OAT' give ^ = ^~ t or tan x = B 2^.. anc | t i ic .

COS &

similar triangles P'd'O and t'aO give cot x = - - . In like manner let the

sin x

student observe the relations when x = APP'P", or an arc between 180 and

270. So also when x = APP'P"P'", or an arc between 270 and 360.

22. COR. 1. The tangent and cotangent of the same angle are

reciprocals of each other ; so also are the secant and cosine, and the

cosecant and the sine. Thus, if tana; 3, cotx ; since cot a; =

1 11

- . If sectf = 2, cosa; = 4-: since sec a; = - , or cosz = - .

tana: cos a; sec a;

23. COR. 2. Sines and cosines cannot exceed' I. Tangents and

cotangents can have any values from fo^oo. Secants and cosecants

can have any values between*! and*-ao . Versed-sines and cover sed-

gines can have any values betiveen and 2. These conclusions' will

readily appear from the definitions, and an inspection of Fig. 4.

SIGNS OF THE TRIGONOMETRICAL FUNCTIONS.

24. Prop. Angles (or arcs) considered as generated from right

to left being called positive * and marked +, those considered as gen-

erated from left to right are to be called negative and marked .

* This ie purely an arbitrary convention. We might equally well reverse It

8

PLANE TRIGONOMETRY.

DEM. This is a direct application of the significance of the 4- and signs.

See Complete School Algebra, pp. 20-23.) Thus, in Pig. 5, if the angle AOP,

considered as generated by the revolution

of a line from the position OA in the direc-

tion of the arrow-head (from right to left),

is called positive and marked + , an angle

generated by the motion of a line from the

position OA in the opposite direction (from

left to right), as the angle AOP" thus gen-

erated, is to be considered negative and

marked . Let it be carefully observed

that it is the assumed direction of the

motion of the generatrix that determines

the sign of the angle (or arc). Two lines

meeting at a common point may be con-

sidered as designating either a -f or a

angle, according to the direction of motion

assumed. Thus the lines OA and OP', Fig.

5, may form the positive angle measured

Fia. 5. by the arc APP', or the negative, salient

angle measured by the negative arc AP'"P"P'. Q. E. D.

25. Prop. Radius being considered as always extending in the

same direction, viz., from the centre toivard the circumference, is

always positive.

26. Prop. The sign of the sine of an angle between and 180

being +, that of an angle between 180 and 360 is .

DEM. In Fig. 5. we observe that the sines of all angles terminating in the 1st

and 2d quadrants, . ., between and 180, may be considered as measured

from the primary diameter AB, upward, while those of angles terminating in the

3d and 4th quadrants, i. e., between 180 and 360, are reckoned downward

(rom the same line; hence, the former being called + , the latter should be ,

is tlie two species are estimated in opposite directions. Q. E. D.

A more elegant conception is to consider the sine as projected upon the diam-

eter vertical to that passing through the origin, as aC ; whence Qd is the sine

of AOP (or arc AP). Now this line evidently is when the angle is ; and as the

angle increases, the sine increases, being generated from upward, and hence

is called + . This is the same conception as we use in the case of the cosine.

Adopting it, we see that sines reckoned from O upward are +, and downward

. Cosines reckoned from to the right, are + , and to the left, .

27. COR. The cosecant of an arc has the same sign as its sine^

eince coseca; = ; and as 1, being the radius, is +, the sign of

sin x

is the same as the sign of sin x.

DEFINITIONS AND FUNDAMENTAL RELATIONS.

9

28. Prop. The sign of the cosine of an angle between and

90, and between 270 and 360, is +, while that of an angle between

90 and 270 is -.

DEM. In Fig. 5, we observe that the cosines of all angles terminating in the

1st and 4th quadrants, may be considered as estimated from the centre towarc.

the right, as OD, OD'"; while correspondingly, the cosines of angles terminating

in the 3d and 3d quadrants will be estimated from the centre toward the left, aa

OD', OD". Hence, by reason of this opposition of direction, the former are

called +, and the latter . Q. E. D.

29. COR. The secant of an angle has the same sign as its cosine,

since these functions are reciprocals of each other. (See #7.)

30. Prop. The sign of the tangent of an angle between and

90, and also between 180 and 270, is + ; while that of an angle

between 90 and 180, and between 270 and 300, is -.

DEM. Since tan x = ?!B_ w hen sin x and cos x have like signs, tan a; is + , by

COS X

the rules of division; and when sin * and cos a- have different signs,* tan a is .

Now, in the 1st and 3d quadrants* the signs of sin a: and cos a are alike, hence

in these quadrants tana; is plus; but in 2d and 4th quadrants sin a: and cos a

have unlike signs, and consequently in these tan x is . Q. E. D.

31. COR. The sign of the cotangent is the same as the sign of the

tangent of the same angle, since cot x = .

32. Prop. Versed-sine and coversed-sine are always +.

DEM. Vers x = 1 - cos x and as cos a; cannot exceed 1, 1 cos a? is al

ways +. In like manner, covers a; = 1 sin x\ and as sin a; cannot exceed 1,

1 sin x is always 4- . Q. E. D.

SCH. 1. It is essential that the law of the signs, as explained above, be well

understood, and the facts fixed in memory. Fig. 6 will aid the student in fixing

the law in the memory. Having this constantly be-

fore the mind, and remembering that tan and cot

are + when sin and cos have like signs, and when

tbey have unlike, and that cos and sec have like

eigns, as also sin and cosec, or, more simply, that

sin 1 1 1

tan = , cot = - , sec = , and cosec = ,

cos tan cos sm

the student cannot fail to know the sign of a func-

tion at a glance.

It will be of service to remember that versed-sine

and coversed-sine, and all the functions of angles

of the 1st quadrant, are -t- ; but that of the other

functions than the versed-sine and coversed-sine, of

angles terminating in the other quadrants, but two are + in each quadrant

ncs.

1.)

"

received

ishers to

ional ap-

) unique,

f calling

asses of

Each volume in the senes is so constructed mat 11 may be used

with equal ease by the youngest and least disciplined who should

be pursuing its theme, and by those who in more mature years

and with more ample preparation enter upon the study.

Sheldon & Company's 2'ext-22oofcs.

Hill's Elements of Rhetoric and Composition

By D. J. HILL, A.M., President Lewisburg University, author

of the Science of Rhetoric. Beginning with the selection of a

theme, this book conducts the learner through every process

of composition, including the accumulation of material, its

arrangement, the choice of words, the construction of sentences,

the variation of expression, the use of figures, the formation of

paragraphs, the preparation of manuscript, and the criticism of

the completed composition.

Hill's Science of Rhetoric

An introduction to the Laws of Effective Discourse. By

D. J. HILL, A.M., President of the University at Lewisburg.

12mo, 300 pages.

This is a thoroughly scientific work on Rhetoric for advanced

classes.

Intellectual Philosophy (ELEMENTS OF). 426 pages

By FRANCIS WAYLAND, late President of Brown Univer-

sity.

The Elements of Moral Science

By FRANCIS WAYLAND, D.D., President of Brown Univer-

sity, and Professor of Moral Philosophy. Fiftieth thousand.

12mo, cloth.

Elements of Political Economy

By FRANCIS WAYLAND, D.D., late President of Brown Uni-

versity. 12mo, cloth, 403 pages.

Recast by AARON L. CHAPIN, D.D., President of Beloit

College.

No text-book on the subject has gained such general accept-

ance, and been so extensively and continuously used, as Dr.

Wayland's. Dr. Chapin has had chiefly in mind the wants of

the class-room, as suggested by an experience of many years.

His aim has been to give in full and proportioned, yet clear

and compact statement, the elements of this important branch

of science, in their latest aspects and applications.

OLNEY'S MATHEMATICAL SERIES.

"ElTEMENTS

TBIGOE"OMETKY

PLANE AND SPHERICAL.

BY EDWARD OLNEY,

MATHEMATICS IN THB UHIVKKSITY Off MICHIGAN

NEW YORK:

SHELDON & COMPANY,

No. 8 MURRAY STREET.

1885-

1

Entered according to Act of Congress in the year 1870. by

SHELDON & COMPANY,

In the Office of the Librarian of Congress at Washington.

PROF, OLNEY'S MATHEMATICAL COURSE,

INTRODUCTION TO ALGEBRA - - -

COMPLETE ALGEBRA

KEY TO COMPLETE ALGEBRA - . - -

UNIVERSITY ALGEBRA

KEY TO UNIVERSITY ALGEBRA -

A VOLUME OF TEST EXAMPLES IN ALGEBRA - - -

ELEMENTS OF GEOMETRY AND TRIGONOMETRY

ELEMENTS OF GEOMETRY AND TRIGONOMETRY, University

Edition

ELEMENTS OF GEOMETRY, separate -

ELEMENTS OF TRIGONOMETRY, separate -

GENERAL GEOMETRY AND CALCULUS -

BELLOWS' TRIGONOMETRY -

PROF, OLNEY'S SERIES OF ARITHMETICS.

PRIMARY ARITHMETIC -

ELEMENTS OF ARITHMETIC - - - -

PRACTICAL ARITHMETIC

SCIENCE OF ARITHMETIC - - - - : - -

E. &. Trig.

CONTENTS.

PART IV. TRIGONOMETRY.

CHAPTER I.

PLANE TRIGONOMETRY.

SECTION I.

DEFINITIONS AND FUNDAMENTAL RELATIONS BETWEEN THE TRIGONOMETRI-

CAL FUNCTIONS OP AN ANGLE (OR ARC). PAGE.

Definitions 1-5

Fundamental Relations of the Trigonometrical Functions of an

Angle 5-7

Signs of the Functions 7-10

Limiting Values of the Functions 10-14

Functions of Negative Arcs 15-16

Circular Functions 10

Exercises 17-20

SECTION II.

RELATIONS BETWEEN THE TRIGONOMETRICAL FUNCTIONS OF DIFFERENT AN-

GLES (OR ARCS).

Functions of the Sum or Difference 20-25

Functions of Double and Half Angles 26

Exercises 26-29

SECTION III.

FORMULAE for rendering Calculable by Logarithms the Algebraic Sum

of Functions 29-30

Exercises 30-31

SECTION IV.

CONSTRUCTION AND USE OF TABLES.

Definitions 31-32

To Compute a Table of Natural Functions 32-33

To Compute a Table of Logarithmic Functions 33-34

Exercises in the Use of the Tables 34-39

Functions of Angles near the Limits of the Quadrant 39-42

Exercises . . 43

800561

IV CONTENTS.

SECTION V.

SOLUTION OF PLANE TRIANGLES. PAGE.

Of Right Angled Triangles 44-45

Exercises and Examples 45-48

Practical Applications 48-49

Of Oblique Angled Plane Triangles 49-51

Exercises 51-55

Oblique Triangles Solved by means of Right Angled Triangles . . . 55-56

Exercises 56-57

functions of the Angles in Terms of the Sides 58-59

Exercises 59-00

Area of Plane Triangles 60-61

Practical Applications 61-64

CHAPTER II.

SPHERICAL TRIGONOMETRY.

INTRODUCTION".

PROJECTION OP SPHERICAL TRIANGLES.

Definitions and Fundamental Propositions 65-66

Projection of Right Angled Spherical Triangles 66-71

Projection of Oblique Angled Spherical Triangles 71-73

SECTION I.

SOLUTION OP RIGHT ANGLED SPHERICAL TRIANGLES.

Definitions , 73-75

Exercises 75-76

Napier's Rules 76-78

Determination of Species 79-81

Exercises in Solution of Right Angled Spherical Triangles 81-84

Quadrantal Triangles 84-85

SECTION II.

OP OBLIQUE ANGLED SPHERICAL TRIANGLES.

To find the Segments of a Side made by a Perpendicular let fall

from the opposite angle 85-86

The Relation of the Sides and Opposite Angles 87

Solution of Oblique Spherical Triangles by Napier's Rules, in

Three Problems 87-90

Exercises .... 90-95

CONTENTS. V

SECTION III.

GENERAL FORMULAE. PAGE.

Angles as Functions of Sides, and Sides as Functions of Angles. 96-100

Gauss's Equations 100-101

Napier's Analogies 102

Exercises in the Use of these Formulae 102-107_

SECTION IV.

Aiea of Spherical Triangles 108-110

Practical Applications of Spherical Trigonometry 110-113

TABLES.

Introduction to the Table of Logarithms 1-10

Table of Logarithms of Numbers 11-28

Table of Natural Sines and Cosines, and of Logarithmic Sines, Cosines,

Tangents, and Cotangents 29-74

Table for Precise Calculation of Functions near their Limits 75-78

Table of Tangents and Cotangents 79-88

PART IV.

TRIGONOMETRY,

CHAPTEE L

PLANE TRIGONOMETRY.

SECTION L

DEFINITIONS AND FUNDAMENTAL RELATIONS BETWEEN THE

TRIGONOMETRICAL FUNCTIONS OF AN ANGLE (OR ARC).

1. Trigonometry is a part of Geometry which has for its sub-

ject-matter, Angles. It is chiefly occupied in presenting a scheme

for measuring and comparing angles, by means of certain auxiliary

lines called Trigonometrical Functions^ investigating the relations

between these functions, and in the solution of triangles by means

of the relations between their sides and the trigonometrical functions

of their angles.

2. Plane Trigonometry treats of plane angles and triangles,

in distinction from Spherical Trigonometry, which treats of spherical

angles and triangles.

3. A Function is a quantity, or a mathematical expression,

conceived as depending upon some other quantity or quantities for

its value.

ILL'S. A man's wages for a given time is a function of the amount received

per day ; or, in general, his wages is a function of both the time of service and

the amount received per day. Again, in the expressions y = 2ax*, y = x* -

2bx + 5,y = 2 log ax, y = a x ,y is a function of a: ; since, the numbers 2, 5, a and b

being considered fixed or constant, the value of y depends upon the value we

assign to x. For a like reason such expressions as \/a a x* , and 3oa; 3 2v/a"j

may be spoken of as functions of a:. Once more, the area of a triangle is a func-

tion of its base and altitude.

4. Angles as Functions of Arcs. Wo have learned in

Geometry (PART II., SEC. VI.), that angles and arcs may be treated

as functions of each other; and that, if the angles be taken at the

I

2 PLANE TRIGONOMETRY.

centre of the same or equal circles, the arcs intercepted nave the

game ratio as the angles themselves, and hence may be taken as then

measures or representatives. For trigonometrical purposes, an angle

is considered as measured by an arc struck with a radius 1, from the

angular point as a centre.

5. A Degree being the -g-J-^ part of the circumference of a circle,

becomes the measure of -fa of a right angle; and, for convenience, it

is customary to speak of such an angle as an angle of one degree, of

four times as large an angle as an angle of four degrees, etc., apply-

ing the term directly to the angle. A small circle written at the

right and a little above a number indicates degrees ().

6. A Minute is fa part of a degree. Minutes are designated by

an accent ('). A Second is -fa part of a minute. Seconds are

indicated by a double accent ("). Smaller divisions of angles (or

arcs) are most conveniently represented as decimals of a second

thougn the designations thirds, fourths, etc., are sometimes met with,

and signify further subdivisions into 60ths. 5 12' 16" 13'" is read,

" 5 degrees, 12 minutes, 16 seconds, and 13 thirds."

ILL'S. In Fig. I AOB is an angle of 35, because the measuring arc ab

contains 35 of the 360 equal parts into which

the circumference whose radius is 0a, could be

divided. In like manner BOC is an angle of 7.

BOC = iAOB = iAOC. Hence, it becomes evi-

dent that we may use the numbers 35, 7, and 42

to represent the respective angles AOB BOC

and AOC, or the corresponding arcs ab, be, and

cue.

7. A Quadrant is an arc of 90,

and is the measure of a right angle;

hence, a right angle is called an angle of

**> * 90. Thus arc ad, Fig. 1, = 90. or angle

AOD = 90.

8. The Complement of an angle or arc is what remains aftei

subtracting the angle or arc from 90. The Supplement of an angle

or arc is what remains after subtracting the angle or arc from 180

ILL'S. In Fig. 1, the angle BOD is the complement of AOB, and the arc 3d

is the complement of arc ab. The complement of 35 is 90 35- 55. Thf

supplement of 35 is 180- 35= 145.

DEFINITIONS AND FUNDAMENTAL RELATIONS. 3

,9. A Quadrant is often represented by ^T, since <n is the semi-

cirou inference when the radius is unity. When this notation is used,

180

the Unit Arc becomes - = 57.29578 nearly, or 57 17' 44 ".8 +,

it

which is an arc equal in length to the radius.

10. For trigonometrical purposes, an angle is conceived as ger

erated by the revolution of a line about the angular point, ana

hence may have any value whatever, not only from to 180, but

from to 360, and even to any number of degrees greater than

360, as 1280, etc. An angle of 45 is generated by of a revolu-

tion, 90 by J of a revolution, 180 by i a revolution, 270 by } of a

revolution, 360 by one revolution, 450 by 1J revolutions, 1280 by

3$ revolutions, etc., etc.

11. In accordance with the conception of an angle as generated

by a revolving line, the measuring arc is considered as originating

at the first position of the revolving line (i. e., with one side of the

angle), and terminating in the line after it has generated the angle

under consideration (i. e., with the other side of the angle), The

first extremity is called the Origin of the arc, and the other the

Termination.

ILL'S. In Fig. 1, let the angle AOB be considered as generated by a line

starting from the position OA, and revolving around the point O, from right to

left,* till it reaches the position OB. Oa being taken as unity, the arc cub is the

measuring arc of the angle AOB ; a is its origin, and b its termination.

12. In the generation of angles by means of a revolving line, the

normal motion is considered to be from right to left, and the quad-

rants are numbered 1st, 2d, 3d, and 4th, in the order in which they

are generated.

13. The Trigonometrical Functions are eight in num-

ber; viz., sine, cosine, tangent, cotangent, secant, cosecant, versed-

sine, and cover sed- sine. These lines are functions of angles, or.

what amounts to the same thing, of arcs considered as measures of

angles, and are the characteristic quantities of trigonometry.

14. The Sine of an angle (or arc) is a perpendicular let fall

from the termination of the measuring arc upon the diameter passing

through the origin of the arc. Thus in Fig. 2, bd is in each case

the sine of the angle AOB, or of the arc axb.

* The pupil will understand that, if he imagines himself standing at the centre of moti .n.

a* the moving body or point passes before him, the distinctions " from right to left." and

14 from left to right," are easily made.

PLANE TKIGONOMETHY.

15. The Trigonometrical Tangent of an angle (or arc)

is a tangent drawn to the measuring arc at its origin, and limited

(a)

CL- A

(c)

by the produced diameter passing through the termination of the

arc. Thus in Fig. %, ac is in each case the tangent of the angle AOB,

or of the arc axb.

16. The Secant of an angle (or arc) is the distance from the

angular point, or centre of the measuring circle, to the extremity of

the tangent of the same angle (or arc). Thus in Fig. 2, Oc is in

each case the secant of the angle AOB, or of the arc axb.

17. The Versed-Sine of an angle (or arc) is the distance

from the foot of the sine of the same angle (or arc) to the origin of

the measuring arc. Thus, in Fig. 2, da is in each case the versed-

sine of the angle AOB, or of the arc axb.

18. The prefix co, in the names of the four trigonometrical func-

tions in which it occurs, is an abbreviation for the word complement.

Thus cosine means complement-sine, i. e., the sine of the comple-

ment; cotangent means tangent of the complement; etc. The co-

sine of 40 is the sine of 90 40, or 50 ; the cosine of 110 is the

sine of 90 110, or 20; the cotangent of 30 is the tangent of

60 ; the cosecant of 200 is the secant of - 110.

j.9. Construction of the Complementary Functions.

Let us now see how the complementary functions are constructed with refer-

uce to their primitives, premising that all arcs in ffy. 3, reckoned from A, are

DEFINITIONS AND FUNDAMENTAL RELATIONS.

to foe reckoned around from right to left in this discussion. 1st. Let AP be

uny arc less than 90 ; then 90' AP = aP is its complement. Now considering

a as the origin and P the termination of

this complementary arc, Pd is its sine, at

its tangent, Ot its secant, and ad its versed -

sine. Hence, Pd, at, Ot, and ad are respect-

ively the cosine, cotangent, cosecant, and

coversed-sine of the arc AP, or the angle

AOP. 2d. Letting APP' be any arc between

90 and 180, its complement is 90 APP'

or aP', the sign signifying that the arc

is reckoned backward from P f to a. But as

the values of the functions will be the same

whether the origin be taken at P' or at a,

we may take a as the origin of this comple-

mentary arc, and P' as its termination,

whence P' d' becomes its sine, at' its tan-

gen t, Ot' its secant, and ad' its versed-sine.

Therefore P'd', at', Of, and ad', are respect-

ively the cosine, cotangent, cosecant, and Fl - 3.

coversed-sine of the arc APP', or the angle AOP'. 3d. In like manner, aP' is

shown to be the complement of arc APP'P" ; and as P"d", at, Ot, and ad!"

are respectively the sine, tangent, secant, and versed-sine of this complement,

they are the corresponding cofunctions of the arc APP'P", or the salient angle

AOP" 4th. In the same way, it appears that P r "d'", at', Of, and ad'" are the

cosine, cotangent, cosecant, and coversed-sine of the arc APP'P" P'", or the

salient angle AOP'". Observe that a, a point on the measuring arc 90 from the

primitive origin, is the origin of all the complementary functions.

SCH. It will readily appear from the figure that the cosine of an angle

(or arc) is always equal to the distance from the foot of tJie sine to the vertex of the

angle (or the centre of the measuring arc). This is the more convenient prac-

tical definition. Thus the cosine of AP is Pd = DO; the cosine of APP' is

P'd 1 = D'O, etc.

20. Notation. Letting x represent any angle (or arc), the

several trigonometrical functions of it are writteL sin a:, cos a;, tana;,

cots, sec a;, cosecz, versa;, and covers x. They are read "sinea,"

" cosine x" " tangent x" " cotangent <B," etc.

FUNDAMENTAL RELATIONS BETWEEN THE TRIGONOMETRICAL

FUNCTIONS OF AN ANGLE (OR ARC).

[Note. These fundamental relations rant he made perfectly familiar They must be

memorized, and be as familiar as the Multiplication Table. The student can do nothing in

trigonometry without them.]

Hf The discussions in this treatise all proceed upon one general

plan ; viz., First obtain the particular propwty of the

6

PLANE T1UGONOMET11Y.

sine and cosine, and from this deduce all the other*

according to the dependencies shown in the follow-

ing proposition.

21. Prop. The Fundamental Relations which the Trigono-

metrical Functiw* *ustain to each other are:

(1) sin 2 x + cos 9 x = 1 ;

. . sin a;

(2) tan x= ;

v ' '

/ox cos a;

(3) cot x = -r

sma

1

(4)

tan x '

(5) seca; = :

cos a;

(6) 1

. ' sin a;'

(7) sec 2 x = 1 '4- tan 8 x ;

(8) cosec'a; = 1 + cot 8 a;

(9) vers x = 1 cos x ;

(10) covers # = 1 sin a:.

(The forms sin a a;, sec 2 a;, etc., signify the square of the sine, the

square of the secant of x, etc., and are read " sine square x," " secant

square x" etc. The student should distinguish between sin a a;, and

sin a; 2 .)

DEM. In Fig. 4, let x represent any arc as AP, less than 90. Then PD = sin *,

OD or Pd = cos#, AT = tana?, OT = seca, at = cotz, OZ = cosecz, AD = versin *,

and ad = co versin x.

FIG. 4.

= ; but op

'

(1). In the right-angled triangle POD

PD 3 + OD 2 OP 2 > 01> sin 2 a; + cos 2 a; = 1,

since OP = radius = 1.

(2). From the similar triangles POD and

TOA,

AT _ PD f> _ sin a;

OA OD' ~cosa;.

(8). From the similar triangles POd and

tOa,

at Pd cos x

= 7r-3, Or COtiC -: .

Oa Od sin x

(4). Multiplying (2) and (3) together,

sin a; cos a; 1

tan#cot2:= : = 1, 01 tan;? =

cos x sin x cot*

(5). From the similar triangles OTA and

OPD,

1

' * ' ~~ COS X

DEFINITIONS AND FUNDAMENTAL RELATIONS.

(6). From the similar triangles Ota and OPd,

Of OP 1

TT- = X-TJ or cosec# = - .

Oa Od' sin a-

(7). From the right-angled triangle OAT,

OT a = OA a -f AT 2 , or sec'z = 1 + tan 1 *.

(8). From the right-angled triangle

0^ = 2 + otf 2 , or cosec' 2 ^ = 1 4- cot a z.

(9). AD = AO - OD, or vers x = 1 - cos x.

(10). ad = aO Od, or covers x 1 sin x.

Thus the fundamental relations of the functions are established for an are

less than 90. But it will readily appear that the relations are the same for any

other arc. For example, let x = AP' be any arc between 90 and 180. Then

the triangle P'D'O gives sin a # + cos 2 z 1, since P'D' = sin #, and OD' = cos x.

The similar triangles P'D'O and OAT' give ^ = ^~ t or tan x = B 2^.. anc | t i ic .

COS &

similar triangles P'd'O and t'aO give cot x = - - . In like manner let the

sin x

student observe the relations when x = APP'P", or an arc between 180 and

270. So also when x = APP'P"P'", or an arc between 270 and 360.

22. COR. 1. The tangent and cotangent of the same angle are

reciprocals of each other ; so also are the secant and cosine, and the

cosecant and the sine. Thus, if tana; 3, cotx ; since cot a; =

1 11

- . If sectf = 2, cosa; = 4-: since sec a; = - , or cosz = - .

tana: cos a; sec a;

23. COR. 2. Sines and cosines cannot exceed' I. Tangents and

cotangents can have any values from fo^oo. Secants and cosecants

can have any values between*! and*-ao . Versed-sines and cover sed-

gines can have any values betiveen and 2. These conclusions' will

readily appear from the definitions, and an inspection of Fig. 4.

SIGNS OF THE TRIGONOMETRICAL FUNCTIONS.

24. Prop. Angles (or arcs) considered as generated from right

to left being called positive * and marked +, those considered as gen-

erated from left to right are to be called negative and marked .

* This ie purely an arbitrary convention. We might equally well reverse It

8

PLANE TRIGONOMETRY.

DEM. This is a direct application of the significance of the 4- and signs.

See Complete School Algebra, pp. 20-23.) Thus, in Pig. 5, if the angle AOP,

considered as generated by the revolution

of a line from the position OA in the direc-

tion of the arrow-head (from right to left),

is called positive and marked + , an angle

generated by the motion of a line from the

position OA in the opposite direction (from

left to right), as the angle AOP" thus gen-

erated, is to be considered negative and

marked . Let it be carefully observed

that it is the assumed direction of the

motion of the generatrix that determines

the sign of the angle (or arc). Two lines

meeting at a common point may be con-

sidered as designating either a -f or a

angle, according to the direction of motion

assumed. Thus the lines OA and OP', Fig.

5, may form the positive angle measured

Fia. 5. by the arc APP', or the negative, salient

angle measured by the negative arc AP'"P"P'. Q. E. D.

25. Prop. Radius being considered as always extending in the

same direction, viz., from the centre toivard the circumference, is

always positive.

26. Prop. The sign of the sine of an angle between and 180

being +, that of an angle between 180 and 360 is .

DEM. In Fig. 5. we observe that the sines of all angles terminating in the 1st

and 2d quadrants, . ., between and 180, may be considered as measured

from the primary diameter AB, upward, while those of angles terminating in the

3d and 4th quadrants, i. e., between 180 and 360, are reckoned downward

(rom the same line; hence, the former being called + , the latter should be ,

is tlie two species are estimated in opposite directions. Q. E. D.

A more elegant conception is to consider the sine as projected upon the diam-

eter vertical to that passing through the origin, as aC ; whence Qd is the sine

of AOP (or arc AP). Now this line evidently is when the angle is ; and as the

angle increases, the sine increases, being generated from upward, and hence

is called + . This is the same conception as we use in the case of the cosine.

Adopting it, we see that sines reckoned from O upward are +, and downward

. Cosines reckoned from to the right, are + , and to the left, .

27. COR. The cosecant of an arc has the same sign as its sine^

eince coseca; = ; and as 1, being the radius, is +, the sign of

sin x

is the same as the sign of sin x.

DEFINITIONS AND FUNDAMENTAL RELATIONS.

9

28. Prop. The sign of the cosine of an angle between and

90, and between 270 and 360, is +, while that of an angle between

90 and 270 is -.

DEM. In Fig. 5, we observe that the cosines of all angles terminating in the

1st and 4th quadrants, may be considered as estimated from the centre towarc.

the right, as OD, OD'"; while correspondingly, the cosines of angles terminating

in the 3d and 3d quadrants will be estimated from the centre toward the left, aa

OD', OD". Hence, by reason of this opposition of direction, the former are

called +, and the latter . Q. E. D.

29. COR. The secant of an angle has the same sign as its cosine,

since these functions are reciprocals of each other. (See #7.)

30. Prop. The sign of the tangent of an angle between and

90, and also between 180 and 270, is + ; while that of an angle

between 90 and 180, and between 270 and 300, is -.

DEM. Since tan x = ?!B_ w hen sin x and cos x have like signs, tan a; is + , by

COS X

the rules of division; and when sin * and cos a- have different signs,* tan a is .

Now, in the 1st and 3d quadrants* the signs of sin a: and cos a are alike, hence

in these quadrants tana; is plus; but in 2d and 4th quadrants sin a: and cos a

have unlike signs, and consequently in these tan x is . Q. E. D.

31. COR. The sign of the cotangent is the same as the sign of the

tangent of the same angle, since cot x = .

32. Prop. Versed-sine and coversed-sine are always +.

DEM. Vers x = 1 - cos x and as cos a; cannot exceed 1, 1 cos a? is al

ways +. In like manner, covers a; = 1 sin x\ and as sin a; cannot exceed 1,

1 sin x is always 4- . Q. E. D.

SCH. 1. It is essential that the law of the signs, as explained above, be well

understood, and the facts fixed in memory. Fig. 6 will aid the student in fixing

the law in the memory. Having this constantly be-

fore the mind, and remembering that tan and cot

are + when sin and cos have like signs, and when

tbey have unlike, and that cos and sec have like

eigns, as also sin and cosec, or, more simply, that

sin 1 1 1

tan = , cot = - , sec = , and cosec = ,

cos tan cos sm

the student cannot fail to know the sign of a func-

tion at a glance.

It will be of service to remember that versed-sine

and coversed-sine, and all the functions of angles

of the 1st quadrant, are -t- ; but that of the other

functions than the versed-sine and coversed-sine, of

angles terminating in the other quadrants, but two are + in each quadrant

Online Library → Edward Olney → Elements of trigonometry, plane and spherical → online text (page 1 of 28)