Online Library → Edward Olney → Elements of trigonometry, plane and spherical → online text (page 2 of 28)

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Fio. 6.

*This is a convenient elliptical form for "an angle whose measuring arc terminates in th

1st quadrant," etc.

10

PLANE TRIGONOMETRY.

Sen. 2. The signs of functions of angles greater than 360" are readily deter

mined by observing in what quadrant the measuring arc terminates. Thus,

sin 570" is , since an arc of 570 terminates in the 3d quadrant. In any given

case, the sign of the function is the same as the sign of the same function of the

remainder left after dividing the arc by 360, or 2X. Thus tan 1180 is the same

as tan 100 ; i. <?., it is . The same may also be said of the value of the func-

tion.

LIMITING} VALUES OF THE TRIGONOMETRICAL FUNCTIONS.

33. Prop. Sin = =FO,

sin

90 = 1, sin 180 = 0,

sin 270 = 1, sin 360= ^0, and the limits within which the

sines of all angles are comprised, are + 1 and I.

DEM. Let the point P be supposed to start from A and move around the

circumference from right to left, and let x represent the angle (or arc) generated.

Now, when P is at A, x = 0, and PD = 0.

Moreover, if we consider the sine P'" D'"

as reaching its limit, by the moving of P'"

from some point in the fourth quadrant to

the origin, the sine of becomes 0, since

what is true of a varying quantity all the way

as it approaches its limit, is assumed true at

the limit. But, if the sine reaches its limit by

the passage of the point P from some point

in the first quadrant to the origin, the sine

of is to be considered as + , since the

function is + as it approaches its limit.

.-. sin = T 0.* As x increases from

to 90 (i. e., as P passes from A to a), PD is

+ and increases from to 1 (-f to + 1).

.'. sine 90 = 1. As # continues to increase

from 90, sin x diminishes and becomes at

Pie. 7. 180. To determine the sign of sin 180, we

notice that it is + when the point P approaches this limit from the second

quadrant, and when it approaches it from the third quadrant, .'. sin 180 = 0.

Conceiving x to go on increasing from 180, sin x appears below AB and is , and

beginning at diminishes (a numerical increase of a negative quantity being

considered an absolute decrease) till at 270 it becomes 1. /. sin 270 = 1.

As x passes from 270 to 360, sin x increases (see above) from 1 to 0. The sign

of this limit is ambiguous, as appears by regarding the limit as reached by the

angle passing from something less than 360 to 360, whence we have ,nd also

as reached by the angle passing back from something greater than 360 to 360,

whence the sign is +. .'. sin 360 = qp 0. Finally, as it is evident that these

values would recur in the same order as the point P passed around again, the

above comprise all real values of sines of angles. Q. E. D.

* It has been customary to disregard the sign of 0, in such a series as m . . 3, - 2,

1, T 0, + 1, + 2, + 3 + m, whereas it should be regarded as ambiguous, as appear*

above.

DEFINITIONS AND FUNDAMENTAL DELATIONS. 11

34. Prop. Cos = 1, cos 90 = =b 0, cos 180 = - 1, cos 270

= =p 0, cos 360 = 1, and the cosines of all angles are comprised

between -f 1 and 1.

DEM. Using the same figure and the same conception as in the last demon-

stration, it is evident that as P approaches A, from either direction, that is as x

approaches 0, the cosine OD approaches to equality with the radius and reaches

It at x = 0, being + in either case. /. cos = 1. As # approaches 90 from a

less value, i. e., from the first quadrant, cos a; is + , and approaching + ; but as

x approaches 90 from some greater value, /. e., from the second quadrant, cos # is

and approaching 0. /. cos 90 = 0. In like manner we observe that as

x increases from 90 to 180, cos x decreases (see DEM. of 33) from to 1, which

it reaches at 180, and this, whether the point is reached in one way or the

other. /. cos 180 = 1. Again, cos 270 T ; since it is if z passes to

270 from some value less than 270, and + 0, if it passes from some value

greater than 270. While x passes through the fourth quadrant, cos x passes

from to + 1, reaching the latter value when x = 360. /. cos 360 = 1. Finally,

as it is evident that the above values would recur in the same order as the gen-

erating point passed around again, this discussion comprises all real values of a

cosine.

3d. Prop. Tan = =p 0, tan 90 = =fc oo, tan 180 = ^ 0,

tan 270 = =b oo , tan 360 = ^ 0, and the tangents of all angles are

Comprised between + oo and oo .

sin 360 T

= cos 30Q o = r- = T 0. From these results it appeal's that the tangent may

Lave any value whatever from +00 to oo ; and as subsequent revolutions of

the generatrix would evidently only repeat these values, these are all the real '

values of a tangent. (Moreover, all real values are comprised between + oo

and - oo ). Q. E. D.

It is easy to observe the direction of the change in the tangent (whether it is

increasing or decreasing) by observing the fraction . As the arc increases in

the first quadrant, the sine increases and the cosine decreases, for both of which

reasons the tangent increases, and hence changes more rapidly than either sine

or cosine. The student should observe the change in each of the four quad-

rants in the same manner.

Sen. These values of the tangent are illustrated by Fig. 7. Thus AT, which

is + , becomes + when P returns to A, or x = 0. Also AT', which is

and may be considered as the tangent of AP'", a negative aic, becomes

whenAP"' = 0. Again, if AP passes to A, AT passes to + *. But, if P'

passes back to a, so that AP' passes to Aa, or 90, AT' passes to oo. /. We

see that tan may be considered T 0, and tan 90 = oo. In like manner th

other limits are illustrated.

12 PLANE TRIGONOMETRY.

36. Prop. Cot = =F oo, cot 90 = 0, corf 180 = =p oo,

corf 270 = 0, corf 360 = qp oo , <m^ rf/^e limits of the cotangent are

+ oo , and oo .

DEM. These values are the reciprocals of the corresponding values of the tan-

gent ; or they may be deduced from the relation cot = -r , in a manner

sin

altogether analogous to those of the tangent. Mg. 7 also affords geometrical

illustrations of them. The student should not fail to deduce and illustrate

them, and also to observe the law of change.

37. Prop. Sec = 1, sec 90 = oo , we 180 = 1, secMO

= ^ oo, sec 360 = 1, and all real values of this function are com-

prised between 1 and oo , and 1 and ^= oo .

DEM. These values are the reciprocals of the corresponding values of the

cosine. The student should obtain them from the cosine, and illustrate them

from Mg. 7, observing the law of change. Thus beginning at OA == 1, which is

the secant of 0, the secant increases till x = 90, when the secant OT becomes

+ oo, if we consider this limit as reached thus ; but oo , if we consider

such an arc as AaP', whose secant is OT', which becomes oo , when the

point P' passes back to a.

Sen. The series which represent the values of secants are, for the first and

second quadrants, + 1, + 2, + 3, -f . . . . + m, oo , - m, . . . . 3, 2,

1 ; for the third and fourth quadrants, 1, 2, 3, . . . . m, T oo ,

+ m, + .... + 3, + 2, + 1, understanding the numbers in these series as rep-

resenting a few terms of series which have an infinite number of terms of all

values extending, in the first case, from + 1 to oo , and thence to 1. It will

be a good exercise for the pupil to write the series representing the values of

each of the trigonometrical functions. Thus, for the sine, we have T 0, + }, + i,

+ $> + 1 + t + fc, + i, 0, for the first and second quadrants, understand-

ing that all values intermediate between those represented are included. For

the second and third quadiants, we have, + 1, + f, + , + 4, 0, , i,

38. Prop.Cosec = ^ oo , cosec 90 = 1, cosec 180 = oo ,

cosec 270 = 1, cosec 360 = ^ oo , and all the real values of this

Unction are comprised between 1 and oo , and I and ^ oo.

DEM. Let the student demonstrate and illustrate as in the preceding article.

Do not neglect to go through the whole in detail ; it is an important and

excellent exercise.

39. Prop. Versin = 0, versm 90 = 1, versin 180 = 2, ver-

sin 270 = 1, versin 360 = 0, and the real limits of the function are

and 2.

DEFINITIONS AND FUNDAMENTAL RELATIONS.

13

DEM. The student will readily deduce these results from the relation vers a

1 cos x. Thus when x = 0, vers = 1 cos 1 1=0, etc.

40. Prop. Covers = 1, covers 90 = 0, covers 180 = 1, covers

270 2, covers 360 = 1, and the limits of the real values of this

function are and 2.

DEM. The student should be able to giye it

41. GENERAL SCHOLIUM. It is important to observe that in the case ol

each of the above functions it changes its sign by passing through or oo . In fact,

it is assumed, in mathematics, that a varying quantity which passes from + to

, does so by passing through or oo . The converse, however, is by no means

time ; viz., that whenever a varying quantity passes through or oo , its sign

necessarily changes.*

TTV

+ 270

* The Co-ordinate Ge-

ometry affords elegant il-

lustrations of the theory

of the change in value and

sign of these functions.

(See Gen. Geom., 23,

etc.) The annexed Fig-

ure represents a curve,

(or as the student may

be disposed to considei

it, a series of curves),

constructed as follows :

On the indefinite line

AE, circumference is

developed (as it were

straightened out), the

origin being at A, an d

AE being the length ol

the circumference. The

Curves mn, m'n', m'W

ire drawn by erecting

at every few degrees

from A, a H ne equal

to the tangent of the

same number of de-

grees, above the line

AE when the tangent is

+ , and below A E when

the tangent is . Thus

AO = 45 in length, and ab = tan 45 ; A^ = 135, and cd = tan 135. Such lines as ab, cd, etc.,

are called ordinates of the curve. The law of change in these ordinates is manifestly the same

as the law of change in tangents. We see that as we pass from to 90, the ordinate (tan-

gent) passes from to + oo. A-. 90 the ordinate (tangent) in both + and , i. ., oo. So also at

270, and at other similar points. A similar device illustrates the changes in the other trigono-

metrical functions. Some may see the propriety of distinguishing QO as + and , who, never-

theless, do not see why it is necessary to make the same distinction in the case of 0. But a

moment's reflection will show that one distinction involves the other, since oo and are mutu-

ally reciprocals of each other.

PIG. 8.

14

PLANE TRIGONOMETRY.

42. SCH. The results of the preceding discussion of the signs and limits (A

the trigonometrical functions, 24 to 41, are exhibited in the annexed

g

.2

OQ

s

s ~

I *

1 5

! j

ll

II

O

fc

3 +

3 +

3 +

TH

3 +

c"

TH

o

TH

+

B

TH

0*

T1

3

3 +

3 +

3 +

3 +

E

o

TH

(M

TH

*

%

o i

T 1

3 +

8

8

3 +

TH

- 1

8

8

H-

3 1

TH

C H-

+

-H

1

8

TH

8

TH

d

-H

3 +

1

3 I

H-

3 1

3 +

%

TH

8

TH

8

+

-H

1

H-

'O

8

O

8

ti

-H

H-

-H

H-

1

3 +

3 1

3 +

3 1

8

8

o

H-

-H

H-

-H

1

8

-H

o

H-

8

-H

o

H-

3 +

- 1

8

3 +

8

H-

-H

H-

H

o

rH

o

TH

I

-H

1

H-

+

o

3 +

3 |

3 1

3 +

^

TH

TH

+

-H

1

H-

TH

o

TH

I

+

-H

1

H-

g

3 +

3 +

3 ;

3 1

8

o

TH

TH

H-

+

-H

1

**

fe

h

<s

i

3

3

3

fc I'M

M

ast*

DEFINITIONS AND FUNDAMENTAL RELATIONS. 15

PKIGONOMETRICAL FUNCTIONS OF NEGATIVE ARCS (OR ANGLES).

43. Prop. Changing tlie sign of an arc (or angle) changes tht

sign of its sine, and consequently of its cosecant; i. e., sin ( x) =

sin x* and cosec ( x) = cosec x.

DEM. 1st, If the angle (or arc) is numerically less than 90, and + , it ends in

the first quadrant, and hence its sine is + ; but, if the angle (or arc) is numeri-

cally lefts than 90 and ,it ends in the fourth quadrant, and hence its sine is .

That is, x being numerically less than 90, sin (x)= sin x. 2d, If x is

between 90 and 180 in numerical value, the arc ends in the second quadrant

when x is 4- , and hence its sine is + ; but when x is , the arc ends in the third

quadrant, whence its sine is . /. In this case, also, sin ( x) = sin x. 3d,

If x is between 180 and 270 in numerical value and +, the arc ends in the

third quadrant, whence sin a? is , ( sin 2); but if the arc is , it ends in the

second quadrant, whence its sign is +, [+ sin( *)]. .*. In this case sin a

= sin ( cr), or sinx = sin( x). 4th, In like manner the student will observe

that sin x = sin ( a 1 ), or sin x = sin ( a?), when x is between 270 and 360.

Moreover, since this order will recur as we pass around again, we learn that

in any case the sign of the sine of a negative angle is the opposite of that of an

equal positive angle. Finally, cosec (- x) = s]]l( _ x) = z^^ =

cosec x, Q. E. D.

44. Prop. Changing the sign of an angle (or arc) does not

cliange the sign of its cosine or secant ; i. e., cos ( x) = cos 2, and

sec ( x) = sec#.

DEM. 1st, When x < 90 and + , the arc ends in the first quadrant, and hence

its cosine is + ; so, also, if x < 90 and , though the arc ends in the fourth

quadrant, its cosine is still + . /. cos ( x) = cos 2. (The student can supply the

other three cases.) Finally, sec ( x) = = = sec x. o. E. D.

cos ( x) cos x

45. Prop. Changing the sign of an angie (or arc) changes

the sign of its tangent, and consequently of its cotangent; i. e.,

tan ( x) = tan x, and cot ( x) = cot x.

* The student should be careful to note the exact meaning of this expression. It is reac

"sine minus x minus sine cc," and means that the sine of a negative angle (or arc) is equa

(numerically) to the sine of the same positive angle (or arc), but has the opposite sign.

16 PLANE TRIGONOMETRY.

DEM. This is an immediate consequence of the fact that changing the sigi.

of the angle (or arc) changes the sign of its sine, but not of its cosine. Tims,

, . sin (a;) sin # sin a; 1

tan ( x) = i = - = tan x. Also, cot ( x) =

cos (-a-) cos a; cos* v ; tan (a-)

= * =- J-^-cotz; or cot(- x) = COS , ( -*> = -ggi. = -ggf

tan x tan x sin ( x) sin x sin a

= cot x.

SCH. The proper sign of a function of a negative angle can always be ascer-

tained by observing in what quadrant the measuring arc ends, in a mannei

altogether similar to that in which the signs of the functions of positive angles

are determined.

CIRCULAR FUNCTIONS.

46. Circular Functions are angles (or arcs) expressed as

functions of sines, cosines, tangents, or other trigonometrical lines.

ILL'S. In the expression sin x, we designate a sine, i. <?., a right line, merely

using the x to tell what sine, as the sine of 20, of 135, etc. But we often wish

to speak of an angle (or arc) which has a particular sine, tangent, or other trigo-

nometrical line. Thus, we say, " the angle (or arc) whose sine is ," " the angle

(or arc) whose tangent is 3," etc. In this form of expression, it is evidently the

angle (or arc) which is the thing mainly thought of; and it is conceived as de-

pendent upon its trigonometrical line.

47. Notation. The circular functions are written sin - l y, cos~ l #,

tan- 1 z, etc. ; and are read "the angle (or arc) whose sine is y" "the

angle whose tangent is z," etc.

ILL'S. The expressions x = sin- 1 ^, and y = sin x, are ultimately equivalent,

since the first is " x = angle whose sine is y" and the second, " y = the sine of a;."

The only difference is, that in the first form the angle (x) is the thing thought of,

and the sine (y) is used merely to tell what angle ; but in the second form, the sine

(y} is the prominent thing, and the angle (x) is used simply to tell what sine. This

mutual relation has caused the circular functions to be called also Inverse Func-

tions.

Sen. This notation is rather an unfortunate one, inasmuch as it is the same

as has been already adopted in the theory of exponents. The student will how-

ever observe that the signification in this instance is altogether different from

the former. Thus, since we write " the square of sine a?," sin a a; ; according to the

theory of exponents, sin- 8 * would be -^ t So also sin -1 a; should mean

sin 3 a? sin x

Now, the former of these expressions would actually signify as indicated (though

it were better to write it (sin a?)- a ), while the latter does not mean at all what

the theory of exponents would make it. Unfortunate as the notation is, it is

probably best to retain it. It, doubtless, was suggested thus : If we have

y =. a'x, we may write x = a*y, so also y = ax, may I e written x = a~ l y.

This affords a parallelism in form, but not in signification.

DEFINITIONS AND FUNDAMENTAL HELATIONS,

EXERCISES.

17

1. What is the complement of 150 21' 13".5 ? What the sup-

plement? Give the complements and also the supplements of

125 15', 283 21' 11", 36 05' 02", and 89 00' 12".

2. How many degrees in the angle (or arc) - ? In J-TT ? In the

urc 2* ? In J* ? In I frr ?

3. How many times is the radius contained in 108? How many

times in 2* ? How many in 460 ? How many in 210 ?

4. Radius being taken as the measure of the arc, by what are 45

represented? By what 90? By what 180? By what 225?

How many degrees does 1 represent, radius being the measure ?

5. Find the length of a degree of the meridian upon a globe of 18

inches diameter.

6. Express 12 22' 13" 11'" 05 lv in , ', ", and decimals of a second,

So also express 53.51 in , ', ", etc.

7. How many degrees, minutes, and seconds in an arc equal to

twice radius ? three times radius ? Show that 27 is equivalent to

ft<r. That 10 to radius 10 ft. = 1.75 ft. What radius gives 1 =,

1 inch ?

8. Draw any angle, and construct its sine, cosine, tangent, or any

other trigonometrical function, and then determine as nearly as prac-

ticable the numerical values of the function by actual measurement,

SOLUTION. Given the angle MON, to find the numerical

value of the tangent as near as practicable by measurement.

Taking any convenient unit, as OA, for a radius, and striking

the arc An, draw AT tangent *o A at A. Now apply OA to

AT and find their ratio (Part I., 36). In this case AT = H

approximately, /. tan MON = 1^.

[NOTE. The student should practice upon such exam-

ples, finding the values of all the functions until the process,

and the meaning of the numerical value of any function

of an angle, are clearly seen.]

9. Construct an angle whose sine is f , i. e., sin~ a -|.

SOLUTION. Let be the required angular point,

and OA one side of the angle. Lay off" from on

OA 3 measures of any convenient length, making

On. Using Oa as a radius, describe the indefinite

arc M. Erect OC perpendicular to OA and take

OC = | of Oa. Through C draw CP parallel to

OA. Finally, draw OB through P. AOB is the

angle required, since Oa being 1, the sine cf

AOB, PD, is i AOB = sin- 1 !.

Fie. 9.

L8 PLANE TBIGONOMETKI.

1C. Construct an angle whose cosine is f. That is, construct

cos" 1 !.

Sua. The construction is the same as in the last example, except that instead

of OC being drawn to limit the arc, a perpendicular is erected to Oa at f (the

second point of division) from 0, and the point P located where this perpendicular

intersects the arc.

11. Construct an angle whose tangent is 2. That is, construct

tan- J 2.

SUG. To construct this at on the line OA, Fig. 10, take any convenient ra-

dius, Oa, and strike the indefinite arc. Then erect at a a tangent, and make it

equal to twice the radius used. Through the extremity of this tangent and

draw a line, aud the angle between this and OA is tan-'S.

12. Construct sec" 1 ^; cot^S; cosec" 1 !^; an obtuse angle sm" 1 ^- ;

tan->(-3).

SUG. To construct sin- 1 !, see Fig. 1. Let OA be one side of the angle, and

the vertex. With any convenient radius draw the semicircumference AaB,

and draw the perpendicular Oa. Bisect this perpendicular, and through the

point of bisection draw a parallel to AB, intersecting the arc in the 2d quad-

rant. Through this intersection draw a line, as OP'. Then AOP' (assuming the

construction as specified, and not as in the figure) = sin" 1 !.

13. Construct the following: tan~'l ; tan~'( 1); tan~'J ;

tan- 1 (-2); tan-'(-i) ; cos-'(-J); sec-'(-2); cosec-'(-3);

versin" 1 } ; versm" 1 !^.

14. From the fundamental relations (21) deduce the following :

sni.r=y / l COS S .T; cos#=Vl sin'a; tanzcota;=l; tan z cos #= sin a;

sin x . cos a; 1

;sma;= ; sina; = . ;cosa;=

tana; cot a;

secz

cot# = coso; coseca;; tana; = ; sin re =

coseca;

secx 1 coseca; cota;

;; vers x = = ; cotcc =

V cosec 2 a; 1 sec a; cosec x

15. Given tan# = f, to find the other trigonometrical functioni

of x.

Results : Sec x = ; cos x = f ; sin x = f ; cosec x = f ; cot

= J ; vers x = \ ; covers x = f .

16. Given sin x = f , to find the other trigonometrical functions

of x.

17. Given sec x = 2, to find the other trigonometrical functions

of x.

18. Given tana = 1, to find the other trigonometrical functionb

rf x.

DEFINITIONS AND FUNDAMENTAL RELATIONS. 19

Results. Cotz = 1; secz = rpA/2~; cosec x = A/2~; sin x

~= db |A/2~; cos? =f= %V% ', versin x = 1 A/2~; coversm * =

i =Fi V2:

[NOTE. Observe closely the signs of the functions in Ex. 18.]

19. In the preceding examples the constructions required have

been limited to angles less than 180, but it is evident that an infi-

nite number of angles (or arcs) according to the more comprehensive

trigonometrical view, correspond to the same function. What angles

or arcs have their tangents each 1 ? What, each 1 ? Construct an

angle between 180 and 270, whose sine is . What other angle

less than 360 has the same sine ?

20. Having given a sine, how many angles less than 180 corre-

spond to it ? Construct the angle or angles less than 180 whose sine

is f . How many angles less than 180 have the same cosine ? tan-

gent ? cotangent ? (In each of the last three cases only one.) Con-

struct y = cos" 1 ^; y cos"^ J). How are these angles related

to each other? Construct # = tan~ a 3; y = tan~ a ( 3). How are

these angles related to each other ? (Restrict the constructions and

questions in this example to angles less than 180.)

21. Given y = sin" 1 ^, show that cosy = A/l~ *> tany =

y =

22. What are trigonometrical functions of 450 ? Of 1350 ? Of

900?

23. Show that the following are true for all integral values of n,

including : sin kn*- ; sin (n + 1) ^ = 1 ; sin (4w + 2) ~ = ;

gin(4n 4- 3) I;cos4rc = l; cos (4w +1) = 0; cos(4

A A & &

= 1 ; cos (4ra + 3) - = ; tan 2| = ; tan (2n + 1) =00 .

66 Z

24. What are the signs of the several trigonometrical function*

of -110? Of -35? Of -500? Of -2000?

25. Prove that sin 30= , cos 30 = fy/3, tan 30= ^3, and

cot 30= A/3, sec 30= | A/3, and cosec 30 =2.

SUG. Observe that the chord of 00 1, and that the sine of 30= i th

chord of 60. Make the figure.

SO PLANE TRIGONOMETRY.

26. Prove that sin 45 = </2, cos 4-5= iy% tan 45= l,cot 45*

= 1, sec 45 V%, and cosec45= A/2-

BUG. Observe that sin 45" cos 45 ; hence sin 2 45 e -f- cos 2 45 = 1, becomes

2sin a 45 = 1.

Sen. The vulues obtained in the last two examples should be retained in the

memory, as they are of frequent use The functions of 30 and of 45 ait

always assumed to be known in any trigonometrical operation.

SECTION II.

RELATIONS BETWEEN THE TRIGONOMETRICAL FUNCTIONS OF

DIFFERENT ANGLES (OR ARCS).

(a) FUNCTIONS OF THE SUM OR DIFFERENCE OF ANGLES

(OR ARCS).

48. J*ro2>. TJie sine of the sum of two angles (or arcs) is equal

to the sine of the first into the cosine of the second, plus the cosine of

the first into the sine of the second. Thus letting x and y represent

any two angles (or arcs),

sin (x + y) = sin x cos y + cos x sin y.

DEM. Let AOB and BOC be the two angles

represented respectively by x and y. Draw the

measuring arc aP', and the sines PD, and P'E of

the angles. AOC = AOB + BOC, is the sum of

the two angles. Draw P'D',the sine of the sum

of the two angles. Then PD = siuaj, P'E sin y,

OD = cos a*, OE = cos y, P'D' = sin (x + y), and

Now sin (a; + y) = P'D' = EF + P'L. But from the similar triangles

and POD, we have -^ = -^ D -, or -L = S -^J? .-. EF = sin x cosy. Also, from

OE OP cosy 1

the similar triangles P'EL and POD, we have ~~ = ^ or

.' P'L = cos x sin y. Substituting these values of EF and P'L, we have

sin (x + y) = since cosy + cos x sin y. Q. E. D.*

* This demonstration may seem defective, since the sum of the angles x and y, as represented

*This is a convenient elliptical form for "an angle whose measuring arc terminates in th

1st quadrant," etc.

10

PLANE TRIGONOMETRY.

Sen. 2. The signs of functions of angles greater than 360" are readily deter

mined by observing in what quadrant the measuring arc terminates. Thus,

sin 570" is , since an arc of 570 terminates in the 3d quadrant. In any given

case, the sign of the function is the same as the sign of the same function of the

remainder left after dividing the arc by 360, or 2X. Thus tan 1180 is the same

as tan 100 ; i. <?., it is . The same may also be said of the value of the func-

tion.

LIMITING} VALUES OF THE TRIGONOMETRICAL FUNCTIONS.

33. Prop. Sin = =FO,

sin

90 = 1, sin 180 = 0,

sin 270 = 1, sin 360= ^0, and the limits within which the

sines of all angles are comprised, are + 1 and I.

DEM. Let the point P be supposed to start from A and move around the

circumference from right to left, and let x represent the angle (or arc) generated.

Now, when P is at A, x = 0, and PD = 0.

Moreover, if we consider the sine P'" D'"

as reaching its limit, by the moving of P'"

from some point in the fourth quadrant to

the origin, the sine of becomes 0, since

what is true of a varying quantity all the way

as it approaches its limit, is assumed true at

the limit. But, if the sine reaches its limit by

the passage of the point P from some point

in the first quadrant to the origin, the sine

of is to be considered as + , since the

function is + as it approaches its limit.

.-. sin = T 0.* As x increases from

to 90 (i. e., as P passes from A to a), PD is

+ and increases from to 1 (-f to + 1).

.'. sine 90 = 1. As # continues to increase

from 90, sin x diminishes and becomes at

Pie. 7. 180. To determine the sign of sin 180, we

notice that it is + when the point P approaches this limit from the second

quadrant, and when it approaches it from the third quadrant, .'. sin 180 = 0.

Conceiving x to go on increasing from 180, sin x appears below AB and is , and

beginning at diminishes (a numerical increase of a negative quantity being

considered an absolute decrease) till at 270 it becomes 1. /. sin 270 = 1.

As x passes from 270 to 360, sin x increases (see above) from 1 to 0. The sign

of this limit is ambiguous, as appears by regarding the limit as reached by the

angle passing from something less than 360 to 360, whence we have ,nd also

as reached by the angle passing back from something greater than 360 to 360,

whence the sign is +. .'. sin 360 = qp 0. Finally, as it is evident that these

values would recur in the same order as the point P passed around again, the

above comprise all real values of sines of angles. Q. E. D.

* It has been customary to disregard the sign of 0, in such a series as m . . 3, - 2,

1, T 0, + 1, + 2, + 3 + m, whereas it should be regarded as ambiguous, as appear*

above.

DEFINITIONS AND FUNDAMENTAL DELATIONS. 11

34. Prop. Cos = 1, cos 90 = =b 0, cos 180 = - 1, cos 270

= =p 0, cos 360 = 1, and the cosines of all angles are comprised

between -f 1 and 1.

DEM. Using the same figure and the same conception as in the last demon-

stration, it is evident that as P approaches A, from either direction, that is as x

approaches 0, the cosine OD approaches to equality with the radius and reaches

It at x = 0, being + in either case. /. cos = 1. As # approaches 90 from a

less value, i. e., from the first quadrant, cos a; is + , and approaching + ; but as

x approaches 90 from some greater value, /. e., from the second quadrant, cos # is

and approaching 0. /. cos 90 = 0. In like manner we observe that as

x increases from 90 to 180, cos x decreases (see DEM. of 33) from to 1, which

it reaches at 180, and this, whether the point is reached in one way or the

other. /. cos 180 = 1. Again, cos 270 T ; since it is if z passes to

270 from some value less than 270, and + 0, if it passes from some value

greater than 270. While x passes through the fourth quadrant, cos x passes

from to + 1, reaching the latter value when x = 360. /. cos 360 = 1. Finally,

as it is evident that the above values would recur in the same order as the gen-

erating point passed around again, this discussion comprises all real values of a

cosine.

3d. Prop. Tan = =p 0, tan 90 = =fc oo, tan 180 = ^ 0,

tan 270 = =b oo , tan 360 = ^ 0, and the tangents of all angles are

Comprised between + oo and oo .

sin 360 T

= cos 30Q o = r- = T 0. From these results it appeal's that the tangent may

Lave any value whatever from +00 to oo ; and as subsequent revolutions of

the generatrix would evidently only repeat these values, these are all the real '

values of a tangent. (Moreover, all real values are comprised between + oo

and - oo ). Q. E. D.

It is easy to observe the direction of the change in the tangent (whether it is

increasing or decreasing) by observing the fraction . As the arc increases in

the first quadrant, the sine increases and the cosine decreases, for both of which

reasons the tangent increases, and hence changes more rapidly than either sine

or cosine. The student should observe the change in each of the four quad-

rants in the same manner.

Sen. These values of the tangent are illustrated by Fig. 7. Thus AT, which

is + , becomes + when P returns to A, or x = 0. Also AT', which is

and may be considered as the tangent of AP'", a negative aic, becomes

whenAP"' = 0. Again, if AP passes to A, AT passes to + *. But, if P'

passes back to a, so that AP' passes to Aa, or 90, AT' passes to oo. /. We

see that tan may be considered T 0, and tan 90 = oo. In like manner th

other limits are illustrated.

12 PLANE TRIGONOMETRY.

36. Prop. Cot = =F oo, cot 90 = 0, corf 180 = =p oo,

corf 270 = 0, corf 360 = qp oo , <m^ rf/^e limits of the cotangent are

+ oo , and oo .

DEM. These values are the reciprocals of the corresponding values of the tan-

gent ; or they may be deduced from the relation cot = -r , in a manner

sin

altogether analogous to those of the tangent. Mg. 7 also affords geometrical

illustrations of them. The student should not fail to deduce and illustrate

them, and also to observe the law of change.

37. Prop. Sec = 1, sec 90 = oo , we 180 = 1, secMO

= ^ oo, sec 360 = 1, and all real values of this function are com-

prised between 1 and oo , and 1 and ^= oo .

DEM. These values are the reciprocals of the corresponding values of the

cosine. The student should obtain them from the cosine, and illustrate them

from Mg. 7, observing the law of change. Thus beginning at OA == 1, which is

the secant of 0, the secant increases till x = 90, when the secant OT becomes

+ oo, if we consider this limit as reached thus ; but oo , if we consider

such an arc as AaP', whose secant is OT', which becomes oo , when the

point P' passes back to a.

Sen. The series which represent the values of secants are, for the first and

second quadrants, + 1, + 2, + 3, -f . . . . + m, oo , - m, . . . . 3, 2,

1 ; for the third and fourth quadrants, 1, 2, 3, . . . . m, T oo ,

+ m, + .... + 3, + 2, + 1, understanding the numbers in these series as rep-

resenting a few terms of series which have an infinite number of terms of all

values extending, in the first case, from + 1 to oo , and thence to 1. It will

be a good exercise for the pupil to write the series representing the values of

each of the trigonometrical functions. Thus, for the sine, we have T 0, + }, + i,

+ $> + 1 + t + fc, + i, 0, for the first and second quadrants, understand-

ing that all values intermediate between those represented are included. For

the second and third quadiants, we have, + 1, + f, + , + 4, 0, , i,

38. Prop.Cosec = ^ oo , cosec 90 = 1, cosec 180 = oo ,

cosec 270 = 1, cosec 360 = ^ oo , and all the real values of this

Unction are comprised between 1 and oo , and I and ^ oo.

DEM. Let the student demonstrate and illustrate as in the preceding article.

Do not neglect to go through the whole in detail ; it is an important and

excellent exercise.

39. Prop. Versin = 0, versm 90 = 1, versin 180 = 2, ver-

sin 270 = 1, versin 360 = 0, and the real limits of the function are

and 2.

DEFINITIONS AND FUNDAMENTAL RELATIONS.

13

DEM. The student will readily deduce these results from the relation vers a

1 cos x. Thus when x = 0, vers = 1 cos 1 1=0, etc.

40. Prop. Covers = 1, covers 90 = 0, covers 180 = 1, covers

270 2, covers 360 = 1, and the limits of the real values of this

function are and 2.

DEM. The student should be able to giye it

41. GENERAL SCHOLIUM. It is important to observe that in the case ol

each of the above functions it changes its sign by passing through or oo . In fact,

it is assumed, in mathematics, that a varying quantity which passes from + to

, does so by passing through or oo . The converse, however, is by no means

time ; viz., that whenever a varying quantity passes through or oo , its sign

necessarily changes.*

TTV

+ 270

* The Co-ordinate Ge-

ometry affords elegant il-

lustrations of the theory

of the change in value and

sign of these functions.

(See Gen. Geom., 23,

etc.) The annexed Fig-

ure represents a curve,

(or as the student may

be disposed to considei

it, a series of curves),

constructed as follows :

On the indefinite line

AE, circumference is

developed (as it were

straightened out), the

origin being at A, an d

AE being the length ol

the circumference. The

Curves mn, m'n', m'W

ire drawn by erecting

at every few degrees

from A, a H ne equal

to the tangent of the

same number of de-

grees, above the line

AE when the tangent is

+ , and below A E when

the tangent is . Thus

AO = 45 in length, and ab = tan 45 ; A^ = 135, and cd = tan 135. Such lines as ab, cd, etc.,

are called ordinates of the curve. The law of change in these ordinates is manifestly the same

as the law of change in tangents. We see that as we pass from to 90, the ordinate (tan-

gent) passes from to + oo. A-. 90 the ordinate (tangent) in both + and , i. ., oo. So also at

270, and at other similar points. A similar device illustrates the changes in the other trigono-

metrical functions. Some may see the propriety of distinguishing QO as + and , who, never-

theless, do not see why it is necessary to make the same distinction in the case of 0. But a

moment's reflection will show that one distinction involves the other, since oo and are mutu-

ally reciprocals of each other.

PIG. 8.

14

PLANE TRIGONOMETRY.

42. SCH. The results of the preceding discussion of the signs and limits (A

the trigonometrical functions, 24 to 41, are exhibited in the annexed

g

.2

OQ

s

s ~

I *

1 5

! j

ll

II

O

fc

3 +

3 +

3 +

TH

3 +

c"

TH

o

TH

+

B

TH

0*

T1

3

3 +

3 +

3 +

3 +

E

o

TH

(M

TH

*

%

o i

T 1

3 +

8

8

3 +

TH

- 1

8

8

H-

3 1

TH

C H-

+

-H

1

8

TH

8

TH

d

-H

3 +

1

3 I

H-

3 1

3 +

%

TH

8

TH

8

+

-H

1

H-

'O

8

O

8

ti

-H

H-

-H

H-

1

3 +

3 1

3 +

3 1

8

8

o

H-

-H

H-

-H

1

8

-H

o

H-

8

-H

o

H-

3 +

- 1

8

3 +

8

H-

-H

H-

H

o

rH

o

TH

I

-H

1

H-

+

o

3 +

3 |

3 1

3 +

^

TH

TH

+

-H

1

H-

TH

o

TH

I

+

-H

1

H-

g

3 +

3 +

3 ;

3 1

8

o

TH

TH

H-

+

-H

1

**

fe

h

<s

i

3

3

3

fc I'M

M

ast*

DEFINITIONS AND FUNDAMENTAL RELATIONS. 15

PKIGONOMETRICAL FUNCTIONS OF NEGATIVE ARCS (OR ANGLES).

43. Prop. Changing tlie sign of an arc (or angle) changes tht

sign of its sine, and consequently of its cosecant; i. e., sin ( x) =

sin x* and cosec ( x) = cosec x.

DEM. 1st, If the angle (or arc) is numerically less than 90, and + , it ends in

the first quadrant, and hence its sine is + ; but, if the angle (or arc) is numeri-

cally lefts than 90 and ,it ends in the fourth quadrant, and hence its sine is .

That is, x being numerically less than 90, sin (x)= sin x. 2d, If x is

between 90 and 180 in numerical value, the arc ends in the second quadrant

when x is 4- , and hence its sine is + ; but when x is , the arc ends in the third

quadrant, whence its sine is . /. In this case, also, sin ( x) = sin x. 3d,

If x is between 180 and 270 in numerical value and +, the arc ends in the

third quadrant, whence sin a? is , ( sin 2); but if the arc is , it ends in the

second quadrant, whence its sign is +, [+ sin( *)]. .*. In this case sin a

= sin ( cr), or sinx = sin( x). 4th, In like manner the student will observe

that sin x = sin ( a 1 ), or sin x = sin ( a?), when x is between 270 and 360.

Moreover, since this order will recur as we pass around again, we learn that

in any case the sign of the sine of a negative angle is the opposite of that of an

equal positive angle. Finally, cosec (- x) = s]]l( _ x) = z^^ =

cosec x, Q. E. D.

44. Prop. Changing the sign of an angle (or arc) does not

cliange the sign of its cosine or secant ; i. e., cos ( x) = cos 2, and

sec ( x) = sec#.

DEM. 1st, When x < 90 and + , the arc ends in the first quadrant, and hence

its cosine is + ; so, also, if x < 90 and , though the arc ends in the fourth

quadrant, its cosine is still + . /. cos ( x) = cos 2. (The student can supply the

other three cases.) Finally, sec ( x) = = = sec x. o. E. D.

cos ( x) cos x

45. Prop. Changing the sign of an angie (or arc) changes

the sign of its tangent, and consequently of its cotangent; i. e.,

tan ( x) = tan x, and cot ( x) = cot x.

* The student should be careful to note the exact meaning of this expression. It is reac

"sine minus x minus sine cc," and means that the sine of a negative angle (or arc) is equa

(numerically) to the sine of the same positive angle (or arc), but has the opposite sign.

16 PLANE TRIGONOMETRY.

DEM. This is an immediate consequence of the fact that changing the sigi.

of the angle (or arc) changes the sign of its sine, but not of its cosine. Tims,

, . sin (a;) sin # sin a; 1

tan ( x) = i = - = tan x. Also, cot ( x) =

cos (-a-) cos a; cos* v ; tan (a-)

= * =- J-^-cotz; or cot(- x) = COS , ( -*> = -ggi. = -ggf

tan x tan x sin ( x) sin x sin a

= cot x.

SCH. The proper sign of a function of a negative angle can always be ascer-

tained by observing in what quadrant the measuring arc ends, in a mannei

altogether similar to that in which the signs of the functions of positive angles

are determined.

CIRCULAR FUNCTIONS.

46. Circular Functions are angles (or arcs) expressed as

functions of sines, cosines, tangents, or other trigonometrical lines.

ILL'S. In the expression sin x, we designate a sine, i. <?., a right line, merely

using the x to tell what sine, as the sine of 20, of 135, etc. But we often wish

to speak of an angle (or arc) which has a particular sine, tangent, or other trigo-

nometrical line. Thus, we say, " the angle (or arc) whose sine is ," " the angle

(or arc) whose tangent is 3," etc. In this form of expression, it is evidently the

angle (or arc) which is the thing mainly thought of; and it is conceived as de-

pendent upon its trigonometrical line.

47. Notation. The circular functions are written sin - l y, cos~ l #,

tan- 1 z, etc. ; and are read "the angle (or arc) whose sine is y" "the

angle whose tangent is z," etc.

ILL'S. The expressions x = sin- 1 ^, and y = sin x, are ultimately equivalent,

since the first is " x = angle whose sine is y" and the second, " y = the sine of a;."

The only difference is, that in the first form the angle (x) is the thing thought of,

and the sine (y) is used merely to tell what angle ; but in the second form, the sine

(y} is the prominent thing, and the angle (x) is used simply to tell what sine. This

mutual relation has caused the circular functions to be called also Inverse Func-

tions.

Sen. This notation is rather an unfortunate one, inasmuch as it is the same

as has been already adopted in the theory of exponents. The student will how-

ever observe that the signification in this instance is altogether different from

the former. Thus, since we write " the square of sine a?," sin a a; ; according to the

theory of exponents, sin- 8 * would be -^ t So also sin -1 a; should mean

sin 3 a? sin x

Now, the former of these expressions would actually signify as indicated (though

it were better to write it (sin a?)- a ), while the latter does not mean at all what

the theory of exponents would make it. Unfortunate as the notation is, it is

probably best to retain it. It, doubtless, was suggested thus : If we have

y =. a'x, we may write x = a*y, so also y = ax, may I e written x = a~ l y.

This affords a parallelism in form, but not in signification.

DEFINITIONS AND FUNDAMENTAL HELATIONS,

EXERCISES.

17

1. What is the complement of 150 21' 13".5 ? What the sup-

plement? Give the complements and also the supplements of

125 15', 283 21' 11", 36 05' 02", and 89 00' 12".

2. How many degrees in the angle (or arc) - ? In J-TT ? In the

urc 2* ? In J* ? In I frr ?

3. How many times is the radius contained in 108? How many

times in 2* ? How many in 460 ? How many in 210 ?

4. Radius being taken as the measure of the arc, by what are 45

represented? By what 90? By what 180? By what 225?

How many degrees does 1 represent, radius being the measure ?

5. Find the length of a degree of the meridian upon a globe of 18

inches diameter.

6. Express 12 22' 13" 11'" 05 lv in , ', ", and decimals of a second,

So also express 53.51 in , ', ", etc.

7. How many degrees, minutes, and seconds in an arc equal to

twice radius ? three times radius ? Show that 27 is equivalent to

ft<r. That 10 to radius 10 ft. = 1.75 ft. What radius gives 1 =,

1 inch ?

8. Draw any angle, and construct its sine, cosine, tangent, or any

other trigonometrical function, and then determine as nearly as prac-

ticable the numerical values of the function by actual measurement,

SOLUTION. Given the angle MON, to find the numerical

value of the tangent as near as practicable by measurement.

Taking any convenient unit, as OA, for a radius, and striking

the arc An, draw AT tangent *o A at A. Now apply OA to

AT and find their ratio (Part I., 36). In this case AT = H

approximately, /. tan MON = 1^.

[NOTE. The student should practice upon such exam-

ples, finding the values of all the functions until the process,

and the meaning of the numerical value of any function

of an angle, are clearly seen.]

9. Construct an angle whose sine is f , i. e., sin~ a -|.

SOLUTION. Let be the required angular point,

and OA one side of the angle. Lay off" from on

OA 3 measures of any convenient length, making

On. Using Oa as a radius, describe the indefinite

arc M. Erect OC perpendicular to OA and take

OC = | of Oa. Through C draw CP parallel to

OA. Finally, draw OB through P. AOB is the

angle required, since Oa being 1, the sine cf

AOB, PD, is i AOB = sin- 1 !.

Fie. 9.

L8 PLANE TBIGONOMETKI.

1C. Construct an angle whose cosine is f. That is, construct

cos" 1 !.

Sua. The construction is the same as in the last example, except that instead

of OC being drawn to limit the arc, a perpendicular is erected to Oa at f (the

second point of division) from 0, and the point P located where this perpendicular

intersects the arc.

11. Construct an angle whose tangent is 2. That is, construct

tan- J 2.

SUG. To construct this at on the line OA, Fig. 10, take any convenient ra-

dius, Oa, and strike the indefinite arc. Then erect at a a tangent, and make it

equal to twice the radius used. Through the extremity of this tangent and

draw a line, aud the angle between this and OA is tan-'S.

12. Construct sec" 1 ^; cot^S; cosec" 1 !^; an obtuse angle sm" 1 ^- ;

tan->(-3).

SUG. To construct sin- 1 !, see Fig. 1. Let OA be one side of the angle, and

the vertex. With any convenient radius draw the semicircumference AaB,

and draw the perpendicular Oa. Bisect this perpendicular, and through the

point of bisection draw a parallel to AB, intersecting the arc in the 2d quad-

rant. Through this intersection draw a line, as OP'. Then AOP' (assuming the

construction as specified, and not as in the figure) = sin" 1 !.

13. Construct the following: tan~'l ; tan~'( 1); tan~'J ;

tan- 1 (-2); tan-'(-i) ; cos-'(-J); sec-'(-2); cosec-'(-3);

versin" 1 } ; versm" 1 !^.

14. From the fundamental relations (21) deduce the following :

sni.r=y / l COS S .T; cos#=Vl sin'a; tanzcota;=l; tan z cos #= sin a;

sin x . cos a; 1

;sma;= ; sina; = . ;cosa;=

tana; cot a;

secz

cot# = coso; coseca;; tana; = ; sin re =

coseca;

secx 1 coseca; cota;

;; vers x = = ; cotcc =

V cosec 2 a; 1 sec a; cosec x

15. Given tan# = f, to find the other trigonometrical functioni

of x.

Results : Sec x = ; cos x = f ; sin x = f ; cosec x = f ; cot

= J ; vers x = \ ; covers x = f .

16. Given sin x = f , to find the other trigonometrical functions

of x.

17. Given sec x = 2, to find the other trigonometrical functions

of x.

18. Given tana = 1, to find the other trigonometrical functionb

rf x.

DEFINITIONS AND FUNDAMENTAL RELATIONS. 19

Results. Cotz = 1; secz = rpA/2~; cosec x = A/2~; sin x

~= db |A/2~; cos? =f= %V% ', versin x = 1 A/2~; coversm * =

i =Fi V2:

[NOTE. Observe closely the signs of the functions in Ex. 18.]

19. In the preceding examples the constructions required have

been limited to angles less than 180, but it is evident that an infi-

nite number of angles (or arcs) according to the more comprehensive

trigonometrical view, correspond to the same function. What angles

or arcs have their tangents each 1 ? What, each 1 ? Construct an

angle between 180 and 270, whose sine is . What other angle

less than 360 has the same sine ?

20. Having given a sine, how many angles less than 180 corre-

spond to it ? Construct the angle or angles less than 180 whose sine

is f . How many angles less than 180 have the same cosine ? tan-

gent ? cotangent ? (In each of the last three cases only one.) Con-

struct y = cos" 1 ^; y cos"^ J). How are these angles related

to each other? Construct # = tan~ a 3; y = tan~ a ( 3). How are

these angles related to each other ? (Restrict the constructions and

questions in this example to angles less than 180.)

21. Given y = sin" 1 ^, show that cosy = A/l~ *> tany =

y =

22. What are trigonometrical functions of 450 ? Of 1350 ? Of

900?

23. Show that the following are true for all integral values of n,

including : sin kn*- ; sin (n + 1) ^ = 1 ; sin (4w + 2) ~ = ;

gin(4n 4- 3) I;cos4rc = l; cos (4w +1) = 0; cos(4

A A & &

= 1 ; cos (4ra + 3) - = ; tan 2| = ; tan (2n + 1) =00 .

66 Z

24. What are the signs of the several trigonometrical function*

of -110? Of -35? Of -500? Of -2000?

25. Prove that sin 30= , cos 30 = fy/3, tan 30= ^3, and

cot 30= A/3, sec 30= | A/3, and cosec 30 =2.

SUG. Observe that the chord of 00 1, and that the sine of 30= i th

chord of 60. Make the figure.

SO PLANE TRIGONOMETRY.

26. Prove that sin 45 = </2, cos 4-5= iy% tan 45= l,cot 45*

= 1, sec 45 V%, and cosec45= A/2-

BUG. Observe that sin 45" cos 45 ; hence sin 2 45 e -f- cos 2 45 = 1, becomes

2sin a 45 = 1.

Sen. The vulues obtained in the last two examples should be retained in the

memory, as they are of frequent use The functions of 30 and of 45 ait

always assumed to be known in any trigonometrical operation.

SECTION II.

RELATIONS BETWEEN THE TRIGONOMETRICAL FUNCTIONS OF

DIFFERENT ANGLES (OR ARCS).

(a) FUNCTIONS OF THE SUM OR DIFFERENCE OF ANGLES

(OR ARCS).

48. J*ro2>. TJie sine of the sum of two angles (or arcs) is equal

to the sine of the first into the cosine of the second, plus the cosine of

the first into the sine of the second. Thus letting x and y represent

any two angles (or arcs),

sin (x + y) = sin x cos y + cos x sin y.

DEM. Let AOB and BOC be the two angles

represented respectively by x and y. Draw the

measuring arc aP', and the sines PD, and P'E of

the angles. AOC = AOB + BOC, is the sum of

the two angles. Draw P'D',the sine of the sum

of the two angles. Then PD = siuaj, P'E sin y,

OD = cos a*, OE = cos y, P'D' = sin (x + y), and

Now sin (a; + y) = P'D' = EF + P'L. But from the similar triangles

and POD, we have -^ = -^ D -, or -L = S -^J? .-. EF = sin x cosy. Also, from

OE OP cosy 1

the similar triangles P'EL and POD, we have ~~ = ^ or

.' P'L = cos x sin y. Substituting these values of EF and P'L, we have

sin (x + y) = since cosy + cos x sin y. Q. E. D.*

* This demonstration may seem defective, since the sum of the angles x and y, as represented

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