Online Library → Edward Olney → Elements of trigonometry, plane and spherical → online text (page 3 of 28)

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ifl the diagram, i less than 90 ; nevertheless, in the General (Analytical) Geometry, we con-

etantly proceed iu a manner entirely analogous ; viz., Arst produce the equation of a locus

from some particular figure, and then make it general in application. The demonstrations in

cases in which the sum of the angles is greater than 90, etc., are similar, and some of them

will be given in the EXERCISES at the close of the section. It is not thought best to cumber tht

purely theoretical part of the subject with such matter.

FUNCTIONS OF THE SUM OR DIFFERENCE OF ANGLES.

COR. Sin(W + x)=cosz. Sin(18Q+x) = -sinx. #^(270 C fa?)

= _ cos x . #m(360 + x) = sin x.

DEM. From the proposition we have,

sin (90' + x) = sin 90 cos x + cos 90 sin x = cos 9 ;

since sin 90 = 1, and cos 90 = 0.

In like manner,

sin (180 + a) = sin 180 cos a; + cos 180 sin x = sin *;

gince sin 180 = 0, and cos 180 = 1.

[The student will readily produce the other forms.]

49. Prof). T7ie sine of the difference between two angles (or arcs)

is equal to the sine of the first into the cosine of the second, minus the

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

represent the angles,

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

DEM. In sin (x + y) = sin* cosy + cos a; siny, substituting y for y, we

have, sin (x y) = sin z cos ( y) * cos x sin ( y] = sin x cos y cos x sin y \

since cos ( y) = cosy, and sin ( y) = siny. (43, 44.)

COR. Sin($Q-x) = cos x. Sin(lSQx) = sinx. Sin(Wx)

= cosx. $w(360 x) = sinx.

DEM. This is simply an application of the proposition, as the preceding

corollary is of its proposition. (The student will make it.) Or we may deduce

the results from the corollary under the preceding proposition by merely sub

stituting x for x. Thus, sin (90 x) = cos ( x) = cos x ; sin (180 x)

sin ( x) ( sin x) = sin x ; etc.

50. Prop. The cosine of the sum of two angles (or arcs) is equal

to the rectangles of their cosines, minus the rectangle of their sines.

Thus, letting x and y represent the angles,

cos (x + y} = cos x cosy sin x sin y.

DEM. Taking the formula sin (x y) = sin x cosy cos x siny, and substi-

tuting 90 x for *, we have, sin (90 x y) = sin (90 x) cos y cos (90*

x) sin y. Now 90 * y = 90 (x + y) ; and sin [90 (x + y)] = the

sine of the complement of (x + y), or cos (x -f y). Also sin (90 x) = cos x

nd cos (90 .r) = sin x. .: Substituting, cos (x -f- y) = cos x cos y sin x sin y.

COR. Co8(W + x)=sinx. Cos(lSO+x)= -cosx. <7o*(270+ ar)

= sin x. ^08(360 + x) = cos x.

DBOI. Apply the proposition.

PLANE TElGONOMETfcY.

51 Prop* The cosine of the difference of two angles (or arcs) Is

equal to the rectangle of their cosines, plus the rectangle of their sines*

Thus, letting x and y represent the angles,

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

DEM. In cos (x+ y) = cos x cos y sin * sin y, substituting y for y, we

have, cos (x y) = cos x cos ( y} sin x sin ( y\ or cos (x y} -= cos x cos y

f sin a: sin y ; since cos ( y) = cos y, and sin ( y) = sin y. [N tice that the

(ast term becomes sinaj( siny), which equals + sin a? sin y.]

COR. Cos(90x)=sinx. Cos(180 x)= cosx. Cos (270 x)

= sin x. Cos (360 x) = cos x.

DEM. Apply the proposition.

62. Prop. TJie tangent of the sum of two angles (or arcs) is

equal to the sum of their tangents, divided by 1 minus the rectangle

of their tangents. Thus, x and y being the angles, we have,

, tan x + tan y

tan (x + y} = -^ r - .

I tan x tan y

_. x sin (x + y) sin x cos y 4- cos x sin y ,,.. . ..

DEM. Tan (a; + y) 7 . : r- 2 . Dividing numer-

cos (x + y} cos x cos y sin x sm y

ator and denominator of the last fraction by cos x cos y* we have, tan (x + y) =

sin x sin y sin x sin y

cos .< cos y cos # cos y tan # + tan v . sin $

. ,SL ., - _ _ sL. since = tana;, etc., and

-sm a sin y sin a; siny 1 tana; tan y cosa?

cos x cos y cos a; cos y

sin # sin y _ sin a sin y

cos a; cos y cos x cos y '

COR. Tan(W+x}= -cotx. Tan(18Q + x)=tanx. Tan(MO

= - co/ a;. 7^^(360 + x) = tan x.

* The three following forms may readily be obtained by dividing respectively by sin x sin y,

lntcoy,andcoBa:iny: viz., tan (x + y) = C ty + ^ , tan te + y) = 1 + COt X tan y -, and

cota; cot y 1 cot a; tan y '

tan (a: + y) = - _tan^~' A " y ne f tliese mav be re( luced to the one given in the prop.

osition, bv substituting in it - for cot, and reducing. Notice that the form in the Prop, is in

terms oJ the tangents. Also observe why dividing by a certain term gives a particular form,

oiul by which to divide to get a required form.

FUNCTIONS OF THE SUM OB DIFFERENCE OF ANGLES. 23

DEM. Tan(90' + 4 = = - = - ** Tan(180'

= tan c

cos (180+ a;) -cos a

3. Prop. 27*e tangent of the difference of two angles (or arcs)

is equal to the difference of their tangents, divided by 1 plus the rect-

angle of their tangents. Thus, x and y being the angles,

tan x tan y

tan (x y) = - - .

v yt l + tan#tan/

sin (x y) sin x cos y cos a? sin y

DEMONSTRATION. Tan (x y) = -. ^ = - . y =.

cos (x y) cos x cos y + sin x sin y

sin x cos y cos a; sin y sin a: _ sin y

cos a cos y cos a; cos y cos a; cosy tan 35 tany

! ^ . W. K. JJ.

cos x cos y sin x siny . sin a? sin y 1 + tan a: tan y

cos ar Cos y cos a; cosy cos cosy

foot-notes to preceding proposition.) This proposition is also readily deduced

from the preceding by substituting in the formula tan (x + y) = D x + an ^.

1 tan x tan $

y for y, and remembering that tan ( y) = tan y.

COR. Zcw(90 -30=e0*as. Tan(lSOx)= tanx.

Tan (360 - ) = tan x.

DEM. Tan (90 - x) = " = = C ol x. Tan (180 - aj) ^

cos (90 a:) sin aj

:r .__ = _ tan ^ Tan(270 o _ x) = -x - cosx

cos (180* - x) -cos x cos (270 - a?) - sin *

sin (360 a) sin a;

cot* Tan(360- x) = = = - tan *.

* These and the kindred formula maybe produced by a direct application of the proposition

tan 90 4- tan x tan 90 1

, tan (OB* + x) = t _ tan 9QO tan g = Z = = ~ C0t * (The reason

dropping tan x and 1 is that they are finite terms connected with infinities, as tan 90 = oo . Or,

tana; 1 tana;

.nee

finite divided by an infinite equals 0. Again, tan (180 + x) = lan 180 1^5, = !

1 tan 180 tan x 1

tan *, since tan 180 = 0, etc.. etc.

24 PLANE TRIGONOMETRY.

54. Prop. The cotangent of the sum of two angles (or arcs) w

equal to the rectangle of their cotangents minus 1, divided by theii

sum. Thus, x and y being the angles,

, , x cot x cot y 1

cot (x + y) = 7 - fc-r .

coty 4- cot a;

cosa; cosy __ sina; siny

DEM Cot/a; + y) = cos fo + y ) - cosa? cosy - sina; siny _ sin.r siny ~ sin* sin^

sin (a? + y) sina; cosy + cosa: sin y sina: cosy cos.esin?/

sina; siny sina; sint

cos x cos y

- x . " 1

sm a; smy cot a: cot y 1 _ . 4 ,.

~- - = - - - . o. E. D. Or we may deduce it thus.

cosy cos a; coty + cotz

sin y sin x

__

1 _ 1 tan x tan y _ cot a; coty _ cot x cot y 1

~~ tan (x + y) ~ tan x + tan y " 1 _J _ cot y + cot x

cot * cot y

Q. E. D.

COR. Cta(90+ a?) = - toz. Cb^(180+ a;) =cot x.

tan x. Cot (360 -h x) = cot x.

DEM. Divide cosine by sine, or take the reciprocals of the corresponding

tangents Thus, cot (90 + x\ = - -^^ - r = - = tan z, etc.

' tan (90 + x) cot a;

55. Prop. The cotangent of the difference of two angles (or

arcs) is equal to the rectangle of their cotangents plus 1, divided by

their difference. Thus, x and y being the angles,

, , , cot x cot y + 1

cot (x y) = - .

cot y cot x

DKM. Substitute y for y, in the preceding formula ; or, divide cos (a; y)

by sin (a? y) and reduce ; or, take the reciprocal of tan (x y), and substitute

for tan.

cat

COR. Cot (90 - x) = tan .r. Cot (180 -a) = - cot x. Cot (270- r]

tan x. Cot (360- x) = cotx.

DEM. Same as above.

Sen. 1. The forrnulm for the secant and cosecant of the sum and of the

difference of two angles (or arcs) are not of sufficient importance to warrant

their introduction here ; some of them will be given in the exercises, as also the

extension of those already given to the case of the sum of three or more angles,

or arcs.

FUNCTIONS OF TIIK SUM OH DIFFERENCE OF ANGLES. 25

Sen. 2. The results reached in this discussion are so important that we will

collect them into a

(A) sin (x + y)

(B) sin (x y)

(C) cos (x + y) -

(D) cos (x y) -

(E) tan (a* + y) =

(F) tan (x - y) -

(G) cot (x + y)

(H) cot (x - y)

TABLE

: sin x cos y + cos x sin y.

sin x cosy cos x sin y.

cos x cos y sin x sin y.

cos x cos y + sin x sin y.

tan # + tan y

1 tana; tany*

tan a? tan y

1 + tan x tan #'

cot a; coty 1

cot y + cot x '

cot x cot y + 1

coty cot &

(D

X

90 a;

90+ x

ISO'S- x

180 4- x

270 x

270+*

360*

sin a?

360*+ac

ine

cos*

cos a;

sin a;

sin x

COS X

COS X

sin x

ooirine

Tangent

sin a;

sin x

cos a;

cos x

sin x

sin a;

cos a;

cos a;

cot*

cot a;

tana;

tan x

cot a;

cot a;

tana;

tan x

cotangent

tana;

tana;

COttt

cot x

tana;

tan x

cot a;

cot x

It will not be found difficult to memorize and extend set (I), if the student

observes, that, when the number of whole quadrants is odd (as 90, 270, etc.),

the function changes name (as from sin to cos, from cos to sin, etc.) ; but, when

the number of whole quadrants is even, the function retains the same name.

The sign of the sine and cosine is readily determined according to fundamental

principles by observing where the arc ends, assuming x < 90. Thus 180+ x

ends in the third quadrant ; hence its sine (which in numerical value is sin x) is

, and its cosine is also . As the signs of the tangent and cotangent of the

same arc are alike, we have only to observe whether the sine and cosine, in any

ejiven case, have like or unlike signs, in order to determine the sign of the tan-

gent and cotangent. For example, what is cot (630 + x) equal to? The nura

\)GT of quadrants being odd (7), the function changes name, and since the arc ends

in the fourth quadrant, its sine is , and its cosine + ; therefore cot (030 + x)

tana;. If in any given case x > 90, determine the character of the function

as above, on the hypothesis x < 90, and then modify the result for the partic-

ular value of x. Thus in the last case, if x was between 90 and 180, its tan-)

gent would be , and for such a value cot (030 + x) = tan a 1 . Or, we may

consider at first where the arc ends, taking into consideration the given value

of x.

26

PLANE TRIGONOMETRY.

() FUNCTIONS OF DOUBLE AND HALF ANGLES.

56. Prop. Letting x represent any angle (or arc),

(K) sin 2x = 2sin x cos x ; (M)

(L) cos 2x = cos a z sin a # =

2cos a x 1, or 1 2sin a x ;

DEM. These results are readily deduced from (A), (C), (E), and (G). Thus, in

sin (x + y) = sin x cos y + cos x sin y y if we make y = x, we have sin 2a? =

sin x cos x + cos x sin x = 2sin a; cos x. (In like manner produce the others.)

1 - tan* o

cot 3 a; 1

(0)

(P)

. Prop. Letting x represent any angle (or arc), we have,

ta**4/E:

f r i +

(Q)

CE)

+ cos a;

1 + cos x

1 cos x

DEM. From (L), 2sin'a; = 1 cos 2a?, or sin x = yi (1 cos 2z). Putting

fe for x t this becomes sin ia; = /y/i (1 cos a;). In like manner, from the same

.ormula (L), 2cos 2 a? = 1 + cos Zx ; whence, cos \x = y^ (1 + cos a:). Again,

tai,=!!Ei?= i /i z .cos f ndco i = i /r+c^;

cos-Ja; r 1 + cos a; tan^a? Y 1 cosx

Sen. The sign of the function in the case of each of these is + if x < 180 ;

but can only be determined by the value of x in any given case.

EXERCISES.

1. Prove from Fig. (a) that sin (x + y) = sin x cos y + cos a; sin y

when x and y are each < 90, but x + y

> 90.

2. Same as in Ex. 1, from (b), when

x < 90, x + y > 90, and < 180, and

y > 90 and < 180.

Sue. In this case, sin (a; + y) - P'D' = P'L -

EF. Iii other respects the demonstration is

identical with the preceding. This gives

+ y) = cos a; sin y sin a; cosy. But the

(a).

FUNCTIONS OF THE SUM Oil DIFFEHENCE OF ANGLES.

27

(6).

sign is accounted for in the general formula,

sin (x + y) = sin x cos y + cos x sin y, by notic-

ing that cosy is , when y > 90 and < 180.

3. Same as in the preceding, when

x > 90, y < 90, and (x + y) < ISO .

BUG. Here sin(# + y) = P'D' = EF - P'L. In

.all other respects the demonstration is identical

with the other cases. The sign in this case

arises froir. x being between 90 and 180,

whence cos x is .

[NOTE. A number of other cases may be

devised, but the rfcove illustrate the varieties.]

4 From Fig. 11, ART. 48, demonstrate

geometrically the formula cos (x + y) =

cos x cos y sin x sin y. The same for

each of the cases in Ex's 1, 2, and 3,

above.

SUG In Fig. 11, cos (x + y) = OD' = OF

L. j~ = X~D> or OF = cos x cos y. - _ = nH> or ^-E = sin a; sin y.

5. Prove geometrically the relation sin (x y)

= sin x cos ?/ cos x sin y.

SUG. Let aP = x, and since y is to be subtracted we

measure it back from P, and y PP'. Now sin (x -y)

- P'D'= EF - P'L.

(d) that cos (x y) =

6. Prove from

cos x cos y + sin x sin

7. Given sin 45= <v/i, and sin 30= J to find sin 75, and sin 15*

Also tan, and cot. Result, sin 75 = .97, sin 15= .26. nearly.

SUG. Use formula, (A .... H).

8. Given sin30=J, to find sine, cosine, tangent, and cotangent^

of 15, 7 30', 3 45', and 1 52' 30". Results, sin 15= .2588, cos 15

= .97, nearly.

SUG. Use the formula in (57). Compare results with those found in the

Table of Natural Sines, etc.

9. Of what angles may the trigonometrical functions he found

from sin 45= JV% hy means of the formulce in (57) ? How ?

28 PLANE TRIGONOMETRY.

10. Prove that sin (x -f y + z) = sin x cos y cos z + cos x sin y cos a

-f cos z cos y sin z sin x sin y sin 2. Also, cos (x + / + 2) =

cosz cos y cos 2 sin x sin # cos z sin x cos y sin z cos # sin y sin 2,

, , tan x + tan y + tan z tan x tan y tan z

Also that tan (z + y + z) = - .

I tan .r tan y tan z tan z tan# tan z

Sue. Sin (a? + y + z) = sin [ (x + y) + z] = sin (x + y) cos 2 + cos (x + y)sin ?.

Sen. Since if (x + y + z) = x, tan (x + y + z) = 0, we have from the last

form, tana; + tany -f tan 2 = tana; tany tan 2; *. ., if a semicircumference be

divided into any three parts, the sum of the tangents of the three parts equals

the products of the tangents of the same.

, x sec a; sec y cosec x cosec y

11. Prove that sec (x + /)= *-

cosec x cosec y sec x sec y

12. Prove that sin 3x = 3sin x 4sin 3 #. Also that cos 3x =

Stan x tan'ic A1

4cos 8 x 3cos a. Also, tan3z = ^ ^- ^ . Also, cot3# =

1 3tan 2 a;

cot 8 x Scot #

3cot a x I

BUGS. Sin 3.c = sin (2x + #) = sin 2x cos a; + cos 2x sin a; = Ssin x cos a; cos x

f (1 2sin a a;) sin x = 2sin a; cos a x + sin a; 2sin 3 x = 2sin a; (1 sin' x) + sin x

2sin* x = 3sin x 4sin' a;.

Tan 3a: = tan (2x + x)= . ten f x TLJ^JH. i n the latter substitute for tan 2x its

1 tan 2a; tan x

value in terms of tan x.

13. Prove that sin 4# = 4 (sin x 2sin 8 ic) cosa;.

2tan x 2

14. Prove that sin x

tan a \x cot \x + tan J

2sin 4a? cos 4a? 2 tan A 2 tan

BUG. Sin x = 2sin ia? cos \x = = - p_ = _

seci secia; sec'Ja; 1+la

COSia:

produce the last form divide numerator and denominator of . - |^- bv

a *

i K r> LI i. i 1 COS iC A , 1 + COS X

15. Prove that tan \x = : - . Also cot x = - : - .

sin x sin a;

SDG. Prom (L), (56*), 2sin a x - 1 cos , and from (K), 2sin \x cos %x = sin ,

Divide the former by the latter.

FORMULA ADAPTED TO LOGARITHMIC COMPUTATION. 29

16. Find the trigonometrical functions of 18.

SOLUTION. Letting x = 18, 2x = 86, and 3 = 54, hence sin 2x = cos 3a>

But sin 2x 2sin x cos x ; and cos 3.c = 4cos* a; 3cos x ; hence 2sin x cos a? ~

4cos s x 3cos , or 2sin x = 4cos a x 3 4 4sin 3 x 3. From which 4sin' a

+ 2sin x \ Solving this quadratic, we have sin x t or sin 18 = *

neglecting the - root, since sin 18 is + . From this, cos 18 = i|/10 + 2^5.

These may be put in approximate decimal fractions.

17. Having given the functions of 18, and 15 (Ex. 8), find those

of 3 ; then of 6, 12, 24, etc.

Compare the results obtained with the values as given in the Table of Natural

Sines, etc., obtaining all the values in decimal fractions.

SECTION IIL

FORMULA FOR RENDERING CALCULABLE BY LOGARITHMS THE

ALGEBRAIC SUM OF TRIGONOMETRICAL FUNCTIONS.

58. Since multiplication, division, involution, and evolution are

the only elementary combinations of number which we can effect by

means of logarithms, if we wish to add or subtract trigonometrical

(or other) quantities, we have first to discover what products,

quotients, powers, or roots, are equivalent to the proposed sums or

differences.

59. Prop. To render sin x sin?/, and coax CQ& y calculable

by logarithms.

SOLUTION. From (55, Sen. 2) we have sin (x + y) = sin x cos y + cos z

bin y, and sin (x y) = sin x cos y cos x sin y. Adding these formulas

sin (x + y) + sin (x y} 2sin x cos y. Now putting x + y x', and x y = y'

whence x = (*' + y'}, and y = i (x f y') ; we have sin' + siny' = 2sm$(x' +V')

cos^(a;' y') ; or. dropping the accents, as the results are general,

(A') sin x + sin y = 2sin $(x + y) cos %(x y).

Again, by subtracting formula B (55, Sen. 2) from formula A, and making

he same substitutions, we have,

( B'l sin * sin y 2cos %(x + y) sin \(x y).

30 PLANE TRIGONOMETRY.

In like manner adding cos (x + y) = cos x cos y sin x sin y, and cos (x y) -

cos # cosy ^ }\nx siny, and making the same substitutions, \ve have,

(C') cos a; + cosy = 2cosi(cc + y) cos \(x - y}.

Finally subtracting formula C (55, Sen. 2) from formula D, and making

the same substitutions, we have,

(DO cos y cos x = 2sin \(x + y) sin i(x y), or

cos x cos y= 2sin \(x + y) sin %(x y).

60. COR.. 1. The sum of the sines of two angles is to their differ-

e",je, as the tangent of one-half the sum of the angles is to the tangent

of one-half their difference.

DEM. Dividing A' by B', we have,

sin x + sin y _ sin j(x + y) cos j(x y) _ sin |(a + y) ^ cos $(x y) _

sin x sin y ~~ cos i(x + y) sin %(x y) ~~ cos %(x + y) sin \(x y) ~~

tan & + y) cot - y) = tou ft, + y) x

6j? COR. 2. TAe difference of the cosines of two angles divided by

their sum is numerically equal to the product of the tangent of one-

half the sum of the two angles into the tangent of one-half their

difference.

DEM. Dividing D' by C', we have,

cos x - cos y _ sin j(x + y) sin j(x y) _ _ sin |(a? + y) sin j(x y) _

cos x + cos y ~~ cos \(x + y) cos i(x y) " cos i(x + y) cos i(x y) ~~

tan KX + y) tan $(x y). Q. E. D. (Observe the opposition in signs.)

62. _PfO&. To render tan x tany calculable ly logarithms.

sin a; sin y sins cosy cosz sin y sin ( y)

DEM. Tan x tan y = - = =

cos x cosy cos x cosy cos a cos #

B. D

EXERCISES.

Let the student deduce the following relations

sin (x + i/}

1. Cot x + cotv = . v T y/ .

sm x sin y

2 cos lx + cos iz v

Sec * - aec j, = + y) si

cosa; cosy

COKfeTUUCTION AND USE OF TRIGONOMETIUCAL TABLES. 31

4. 1 + cos x = 2cos 3 ^. (See J7.)

5. 1 cos a; =

6. sina; + sm ^ = tan 4(3 + y). (Divide A' by 0', SO.)

cos a + cos

, sm x sin y , , .

7. - -= tan4(z-y).

cos x + cos y

t sin x 4- sin y

8. - 2. _ C ot 4(z y).

cos x cos y

rins-riny^

cos x cos y

SUCTION IV.

CONSTRUCTION AND USE OF TRIGONOMETRICAL TABLES.

[NOTE. In order to read this and the subsequent sections, the student needa

a knowledge of the nature of logarithms, and the method of using common

logarithmic tables. If he is familiar with the last chapter in THE COMPLETE

SCHOOL ALGEBRA of this series, he is prepared to go on. If he has not this

knowledge, he should read the introduction preceding the table of Logarithms

before reading this section.]

63. A Table of Trigonometrical Functions is a table

containing the values of these functions corresponding to angles of

all different values. In consequence of the incommensurability of

an arc and its functions, these results can be given only approxi-

mately; yet it is possible to attain any degree of accuracy which

practical science requires.

G4i* There are two tables of trigonometrical functions in common

use, the Table of Natural Functions, and the Table of Logarithmic

Functions.

65* A Table of Natural Trigonometrical Functions

is a table in which are written the values of these functions for

angles of various values, the radius of the circle being taken as the

measuring unit, and the function being expressed in natural num-

bers extended to as many decimal places as the proposed degree of

accuracy requires.

66. A Table of Logarithmic Trigonometrical Func-

tions is the same as a table of natural functions, except that the

logarithms of the values of the functions are written instead of the

functions themselves, and to avoid tbe frequent occurrence of nes;a-

32 PLANE TRIGONOMETRY.

tive characteristics, the characteristic of each logarithm is increased

by 10. For example, sines and cosines being always less than unity,

except at the limit (33), and tangents of angles less than 45 and

cotangents of angles greater than 45 being also less than unity, the

logarithms of all such functions have negative characteristics. To

obviate the necessity of writing these with their sign, the charac-

teristic of each logarithm is increased by 10.

67. JProb. To compute a table of natural trigonometrical func-

tions for every degree and minute of the quadrant.

SOLUTION. It is evident that an arc is longer than its sine, but that this

disparity diminishes as the arc grows less. Thus, in a circle whose radius is

1 inch, the length of the sine of an arc of 1 would not differ appreciably from

the arc. Much less should we be able to distinguish between the sine of 1' and

the arc. Now, since when the radius is 1, a seinicircumference = it = 3.1415926,

and also = 180, or 180 X 60 10800', we have the length of an arc of 1' =

' = 0.0002908882 approximately. Assuming this as the sine of 1', we

obtain the cosine thus,

cos V =Vl-sin a l' =V(l + sinl')X(l-sinl') = v/1.0002908882 X .9997091118

= 0.9999999577.

Having thus obtained sufficiently accurate values of sin 1' and cos 1', we can

continue the operation as follows : from the formulae sin (x + y} + sin (x y) =

2 sin x cos y, and cos (x + y) + cos (x y} = 2 cos x cos y, we have

sin (x + y) = 2 sin x cos y sin (x y\

cos (x + y) 2 cos x cos y cos (x y).

Now letting y remain constantly equal to 1', and letting x take successively

the values 1', 2', 3', etc., we have

, ( sin 2' = 2 cos V sin 1' - sin 0' = 0.0005817764

1 K

' ( cos 2' = 2 cos 1' cos V - cos 0' = 0.9999998308

j sin 3' = 2 cos 1' sin 2' - sin 1' = 0.0008726646

1 cos 3' = 2 cos 1' cos 2' - cos V = 0.9999996193

sin 4' = 2 cos 1' sin 3' sin 2' = 0.0011635526

_

' I cos 4' = 2 cos V cos 3' - cos 2' = 0.9999993232

j sin 5' = 2 cos 1' sin 4' - sin 3' = 0.0014544407

4 ' \ cos 5' = 2 cos 1' cos 4' - cos 3' = 0.9999989425

etc., etc.

These operations present no difficulties except the labor of performing the

numerical operations.

Of course 60 operations are required for every degree, and far 30, 1800. Bui

having computed the shies and cosines for every degree and minute up to 30

CONSTRUCTION AND USE OF TRIGONOMETRICAL TABLES. 33

we can complete the work by simple subtraction of values already found. For

example, letting x = 30, tbe first formula used above becomes

sin (30 + y) = cos y - sin (30 - y\

and from cos (x + y) cos (x y} = 2 sin x sin y, we have

cos (30+ y) = cos (30 - y} - sin y

Now making y successively = 1', 2', 3', etc., these give

j sin 30' 1' = cos 1' sin 29 59'

( cos 30 V = cos 29 59' sin 1'

j sin 30 2' = cos 2' - sin 29 58'

( cos 30 2' = cos 29 58' - sin 2'

j sin 30 3' = cos 3' - sin 29 57

\ cos 30 3' = cos 29 57' - sin 3'

etc., etc.

All of these values which occur in the second members having been deter-

mined in reaching sin 30 and cos 30, those in the first members can be found

by performing the requisite subtractions.

Proceeding in this way till we reach 45, the numerical values of all sines and

cosines become known, since the sine of any angle between 45 and 90", being

the cosine of the complementary angle, will have been computed in reaching

45. And so also the cosines of angles between 45 and 90 will have been

computed as sines of the complementary angles below 45.

The sines and cosines being computed, the corresponding tangents, cotan-

gents, and, if need be, the secants, cosecants, versed-sines, and coversed-

sin x 1 cos x

sines, can be calculated from the relations tana; = , cot a = or - ,

cos x 1 tan x sin x

sec a = , coseca; = . , versa; = 1 cos a;, and covers x = 1 sin a?.

etantly proceed iu a manner entirely analogous ; viz., Arst produce the equation of a locus

from some particular figure, and then make it general in application. The demonstrations in

cases in which the sum of the angles is greater than 90, etc., are similar, and some of them

will be given in the EXERCISES at the close of the section. It is not thought best to cumber tht

purely theoretical part of the subject with such matter.

FUNCTIONS OF THE SUM OR DIFFERENCE OF ANGLES.

COR. Sin(W + x)=cosz. Sin(18Q+x) = -sinx. #^(270 C fa?)

= _ cos x . #m(360 + x) = sin x.

DEM. From the proposition we have,

sin (90' + x) = sin 90 cos x + cos 90 sin x = cos 9 ;

since sin 90 = 1, and cos 90 = 0.

In like manner,

sin (180 + a) = sin 180 cos a; + cos 180 sin x = sin *;

gince sin 180 = 0, and cos 180 = 1.

[The student will readily produce the other forms.]

49. Prof). T7ie sine of the difference between two angles (or arcs)

is equal to the sine of the first into the cosine of the second, minus the

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

represent the angles,

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

DEM. In sin (x + y) = sin* cosy + cos a; siny, substituting y for y, we

have, sin (x y) = sin z cos ( y) * cos x sin ( y] = sin x cos y cos x sin y \

since cos ( y) = cosy, and sin ( y) = siny. (43, 44.)

COR. Sin($Q-x) = cos x. Sin(lSQx) = sinx. Sin(Wx)

= cosx. $w(360 x) = sinx.

DEM. This is simply an application of the proposition, as the preceding

corollary is of its proposition. (The student will make it.) Or we may deduce

the results from the corollary under the preceding proposition by merely sub

stituting x for x. Thus, sin (90 x) = cos ( x) = cos x ; sin (180 x)

sin ( x) ( sin x) = sin x ; etc.

50. Prop. The cosine of the sum of two angles (or arcs) is equal

to the rectangles of their cosines, minus the rectangle of their sines.

Thus, letting x and y represent the angles,

cos (x + y} = cos x cosy sin x sin y.

DEM. Taking the formula sin (x y) = sin x cosy cos x siny, and substi-

tuting 90 x for *, we have, sin (90 x y) = sin (90 x) cos y cos (90*

x) sin y. Now 90 * y = 90 (x + y) ; and sin [90 (x + y)] = the

sine of the complement of (x + y), or cos (x -f y). Also sin (90 x) = cos x

nd cos (90 .r) = sin x. .: Substituting, cos (x -f- y) = cos x cos y sin x sin y.

COR. Co8(W + x)=sinx. Cos(lSO+x)= -cosx. <7o*(270+ ar)

= sin x. ^08(360 + x) = cos x.

DBOI. Apply the proposition.

PLANE TElGONOMETfcY.

51 Prop* The cosine of the difference of two angles (or arcs) Is

equal to the rectangle of their cosines, plus the rectangle of their sines*

Thus, letting x and y represent the angles,

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

DEM. In cos (x+ y) = cos x cos y sin * sin y, substituting y for y, we

have, cos (x y) = cos x cos ( y} sin x sin ( y\ or cos (x y} -= cos x cos y

f sin a: sin y ; since cos ( y) = cos y, and sin ( y) = sin y. [N tice that the

(ast term becomes sinaj( siny), which equals + sin a? sin y.]

COR. Cos(90x)=sinx. Cos(180 x)= cosx. Cos (270 x)

= sin x. Cos (360 x) = cos x.

DEM. Apply the proposition.

62. Prop. TJie tangent of the sum of two angles (or arcs) is

equal to the sum of their tangents, divided by 1 minus the rectangle

of their tangents. Thus, x and y being the angles, we have,

, tan x + tan y

tan (x + y} = -^ r - .

I tan x tan y

_. x sin (x + y) sin x cos y 4- cos x sin y ,,.. . ..

DEM. Tan (a; + y) 7 . : r- 2 . Dividing numer-

cos (x + y} cos x cos y sin x sm y

ator and denominator of the last fraction by cos x cos y* we have, tan (x + y) =

sin x sin y sin x sin y

cos .< cos y cos # cos y tan # + tan v . sin $

. ,SL ., - _ _ sL. since = tana;, etc., and

-sm a sin y sin a; siny 1 tana; tan y cosa?

cos x cos y cos a; cos y

sin # sin y _ sin a sin y

cos a; cos y cos x cos y '

COR. Tan(W+x}= -cotx. Tan(18Q + x)=tanx. Tan(MO

= - co/ a;. 7^^(360 + x) = tan x.

* The three following forms may readily be obtained by dividing respectively by sin x sin y,

lntcoy,andcoBa:iny: viz., tan (x + y) = C ty + ^ , tan te + y) = 1 + COt X tan y -, and

cota; cot y 1 cot a; tan y '

tan (a: + y) = - _tan^~' A " y ne f tliese mav be re( luced to the one given in the prop.

osition, bv substituting in it - for cot, and reducing. Notice that the form in the Prop, is in

terms oJ the tangents. Also observe why dividing by a certain term gives a particular form,

oiul by which to divide to get a required form.

FUNCTIONS OF THE SUM OB DIFFERENCE OF ANGLES. 23

DEM. Tan(90' + 4 = = - = - ** Tan(180'

= tan c

cos (180+ a;) -cos a

3. Prop. 27*e tangent of the difference of two angles (or arcs)

is equal to the difference of their tangents, divided by 1 plus the rect-

angle of their tangents. Thus, x and y being the angles,

tan x tan y

tan (x y) = - - .

v yt l + tan#tan/

sin (x y) sin x cos y cos a? sin y

DEMONSTRATION. Tan (x y) = -. ^ = - . y =.

cos (x y) cos x cos y + sin x sin y

sin x cos y cos a; sin y sin a: _ sin y

cos a cos y cos a; cos y cos a; cosy tan 35 tany

! ^ . W. K. JJ.

cos x cos y sin x siny . sin a? sin y 1 + tan a: tan y

cos ar Cos y cos a; cosy cos cosy

foot-notes to preceding proposition.) This proposition is also readily deduced

from the preceding by substituting in the formula tan (x + y) = D x + an ^.

1 tan x tan $

y for y, and remembering that tan ( y) = tan y.

COR. Zcw(90 -30=e0*as. Tan(lSOx)= tanx.

Tan (360 - ) = tan x.

DEM. Tan (90 - x) = " = = C ol x. Tan (180 - aj) ^

cos (90 a:) sin aj

:r .__ = _ tan ^ Tan(270 o _ x) = -x - cosx

cos (180* - x) -cos x cos (270 - a?) - sin *

sin (360 a) sin a;

cot* Tan(360- x) = = = - tan *.

* These and the kindred formula maybe produced by a direct application of the proposition

tan 90 4- tan x tan 90 1

, tan (OB* + x) = t _ tan 9QO tan g = Z = = ~ C0t * (The reason

dropping tan x and 1 is that they are finite terms connected with infinities, as tan 90 = oo . Or,

tana; 1 tana;

.nee

finite divided by an infinite equals 0. Again, tan (180 + x) = lan 180 1^5, = !

1 tan 180 tan x 1

tan *, since tan 180 = 0, etc.. etc.

24 PLANE TRIGONOMETRY.

54. Prop. The cotangent of the sum of two angles (or arcs) w

equal to the rectangle of their cotangents minus 1, divided by theii

sum. Thus, x and y being the angles,

, , x cot x cot y 1

cot (x + y) = 7 - fc-r .

coty 4- cot a;

cosa; cosy __ sina; siny

DEM Cot/a; + y) = cos fo + y ) - cosa? cosy - sina; siny _ sin.r siny ~ sin* sin^

sin (a? + y) sina; cosy + cosa: sin y sina: cosy cos.esin?/

sina; siny sina; sint

cos x cos y

- x . " 1

sm a; smy cot a: cot y 1 _ . 4 ,.

~- - = - - - . o. E. D. Or we may deduce it thus.

cosy cos a; coty + cotz

sin y sin x

__

1 _ 1 tan x tan y _ cot a; coty _ cot x cot y 1

~~ tan (x + y) ~ tan x + tan y " 1 _J _ cot y + cot x

cot * cot y

Q. E. D.

COR. Cta(90+ a?) = - toz. Cb^(180+ a;) =cot x.

tan x. Cot (360 -h x) = cot x.

DEM. Divide cosine by sine, or take the reciprocals of the corresponding

tangents Thus, cot (90 + x\ = - -^^ - r = - = tan z, etc.

' tan (90 + x) cot a;

55. Prop. The cotangent of the difference of two angles (or

arcs) is equal to the rectangle of their cotangents plus 1, divided by

their difference. Thus, x and y being the angles,

, , , cot x cot y + 1

cot (x y) = - .

cot y cot x

DKM. Substitute y for y, in the preceding formula ; or, divide cos (a; y)

by sin (a? y) and reduce ; or, take the reciprocal of tan (x y), and substitute

for tan.

cat

COR. Cot (90 - x) = tan .r. Cot (180 -a) = - cot x. Cot (270- r]

tan x. Cot (360- x) = cotx.

DEM. Same as above.

Sen. 1. The forrnulm for the secant and cosecant of the sum and of the

difference of two angles (or arcs) are not of sufficient importance to warrant

their introduction here ; some of them will be given in the exercises, as also the

extension of those already given to the case of the sum of three or more angles,

or arcs.

FUNCTIONS OF TIIK SUM OH DIFFERENCE OF ANGLES. 25

Sen. 2. The results reached in this discussion are so important that we will

collect them into a

(A) sin (x + y)

(B) sin (x y)

(C) cos (x + y) -

(D) cos (x y) -

(E) tan (a* + y) =

(F) tan (x - y) -

(G) cot (x + y)

(H) cot (x - y)

TABLE

: sin x cos y + cos x sin y.

sin x cosy cos x sin y.

cos x cos y sin x sin y.

cos x cos y + sin x sin y.

tan # + tan y

1 tana; tany*

tan a? tan y

1 + tan x tan #'

cot a; coty 1

cot y + cot x '

cot x cot y + 1

coty cot &

(D

X

90 a;

90+ x

ISO'S- x

180 4- x

270 x

270+*

360*

sin a?

360*+ac

ine

cos*

cos a;

sin a;

sin x

COS X

COS X

sin x

ooirine

Tangent

sin a;

sin x

cos a;

cos x

sin x

sin a;

cos a;

cos a;

cot*

cot a;

tana;

tan x

cot a;

cot a;

tana;

tan x

cotangent

tana;

tana;

COttt

cot x

tana;

tan x

cot a;

cot x

It will not be found difficult to memorize and extend set (I), if the student

observes, that, when the number of whole quadrants is odd (as 90, 270, etc.),

the function changes name (as from sin to cos, from cos to sin, etc.) ; but, when

the number of whole quadrants is even, the function retains the same name.

The sign of the sine and cosine is readily determined according to fundamental

principles by observing where the arc ends, assuming x < 90. Thus 180+ x

ends in the third quadrant ; hence its sine (which in numerical value is sin x) is

, and its cosine is also . As the signs of the tangent and cotangent of the

same arc are alike, we have only to observe whether the sine and cosine, in any

ejiven case, have like or unlike signs, in order to determine the sign of the tan-

gent and cotangent. For example, what is cot (630 + x) equal to? The nura

\)GT of quadrants being odd (7), the function changes name, and since the arc ends

in the fourth quadrant, its sine is , and its cosine + ; therefore cot (030 + x)

tana;. If in any given case x > 90, determine the character of the function

as above, on the hypothesis x < 90, and then modify the result for the partic-

ular value of x. Thus in the last case, if x was between 90 and 180, its tan-)

gent would be , and for such a value cot (030 + x) = tan a 1 . Or, we may

consider at first where the arc ends, taking into consideration the given value

of x.

26

PLANE TRIGONOMETRY.

() FUNCTIONS OF DOUBLE AND HALF ANGLES.

56. Prop. Letting x represent any angle (or arc),

(K) sin 2x = 2sin x cos x ; (M)

(L) cos 2x = cos a z sin a # =

2cos a x 1, or 1 2sin a x ;

DEM. These results are readily deduced from (A), (C), (E), and (G). Thus, in

sin (x + y) = sin x cos y + cos x sin y y if we make y = x, we have sin 2a? =

sin x cos x + cos x sin x = 2sin a; cos x. (In like manner produce the others.)

1 - tan* o

cot 3 a; 1

(0)

(P)

. Prop. Letting x represent any angle (or arc), we have,

ta**4/E:

f r i +

(Q)

CE)

+ cos a;

1 + cos x

1 cos x

DEM. From (L), 2sin'a; = 1 cos 2a?, or sin x = yi (1 cos 2z). Putting

fe for x t this becomes sin ia; = /y/i (1 cos a;). In like manner, from the same

.ormula (L), 2cos 2 a? = 1 + cos Zx ; whence, cos \x = y^ (1 + cos a:). Again,

tai,=!!Ei?= i /i z .cos f ndco i = i /r+c^;

cos-Ja; r 1 + cos a; tan^a? Y 1 cosx

Sen. The sign of the function in the case of each of these is + if x < 180 ;

but can only be determined by the value of x in any given case.

EXERCISES.

1. Prove from Fig. (a) that sin (x + y) = sin x cos y + cos a; sin y

when x and y are each < 90, but x + y

> 90.

2. Same as in Ex. 1, from (b), when

x < 90, x + y > 90, and < 180, and

y > 90 and < 180.

Sue. In this case, sin (a; + y) - P'D' = P'L -

EF. Iii other respects the demonstration is

identical with the preceding. This gives

+ y) = cos a; sin y sin a; cosy. But the

(a).

FUNCTIONS OF THE SUM Oil DIFFEHENCE OF ANGLES.

27

(6).

sign is accounted for in the general formula,

sin (x + y) = sin x cos y + cos x sin y, by notic-

ing that cosy is , when y > 90 and < 180.

3. Same as in the preceding, when

x > 90, y < 90, and (x + y) < ISO .

BUG. Here sin(# + y) = P'D' = EF - P'L. In

.all other respects the demonstration is identical

with the other cases. The sign in this case

arises froir. x being between 90 and 180,

whence cos x is .

[NOTE. A number of other cases may be

devised, but the rfcove illustrate the varieties.]

4 From Fig. 11, ART. 48, demonstrate

geometrically the formula cos (x + y) =

cos x cos y sin x sin y. The same for

each of the cases in Ex's 1, 2, and 3,

above.

SUG In Fig. 11, cos (x + y) = OD' = OF

L. j~ = X~D> or OF = cos x cos y. - _ = nH> or ^-E = sin a; sin y.

5. Prove geometrically the relation sin (x y)

= sin x cos ?/ cos x sin y.

SUG. Let aP = x, and since y is to be subtracted we

measure it back from P, and y PP'. Now sin (x -y)

- P'D'= EF - P'L.

(d) that cos (x y) =

6. Prove from

cos x cos y + sin x sin

7. Given sin 45= <v/i, and sin 30= J to find sin 75, and sin 15*

Also tan, and cot. Result, sin 75 = .97, sin 15= .26. nearly.

SUG. Use formula, (A .... H).

8. Given sin30=J, to find sine, cosine, tangent, and cotangent^

of 15, 7 30', 3 45', and 1 52' 30". Results, sin 15= .2588, cos 15

= .97, nearly.

SUG. Use the formula in (57). Compare results with those found in the

Table of Natural Sines, etc.

9. Of what angles may the trigonometrical functions he found

from sin 45= JV% hy means of the formulce in (57) ? How ?

28 PLANE TRIGONOMETRY.

10. Prove that sin (x -f y + z) = sin x cos y cos z + cos x sin y cos a

-f cos z cos y sin z sin x sin y sin 2. Also, cos (x + / + 2) =

cosz cos y cos 2 sin x sin # cos z sin x cos y sin z cos # sin y sin 2,

, , tan x + tan y + tan z tan x tan y tan z

Also that tan (z + y + z) = - .

I tan .r tan y tan z tan z tan# tan z

Sue. Sin (a? + y + z) = sin [ (x + y) + z] = sin (x + y) cos 2 + cos (x + y)sin ?.

Sen. Since if (x + y + z) = x, tan (x + y + z) = 0, we have from the last

form, tana; + tany -f tan 2 = tana; tany tan 2; *. ., if a semicircumference be

divided into any three parts, the sum of the tangents of the three parts equals

the products of the tangents of the same.

, x sec a; sec y cosec x cosec y

11. Prove that sec (x + /)= *-

cosec x cosec y sec x sec y

12. Prove that sin 3x = 3sin x 4sin 3 #. Also that cos 3x =

Stan x tan'ic A1

4cos 8 x 3cos a. Also, tan3z = ^ ^- ^ . Also, cot3# =

1 3tan 2 a;

cot 8 x Scot #

3cot a x I

BUGS. Sin 3.c = sin (2x + #) = sin 2x cos a; + cos 2x sin a; = Ssin x cos a; cos x

f (1 2sin a a;) sin x = 2sin a; cos a x + sin a; 2sin 3 x = 2sin a; (1 sin' x) + sin x

2sin* x = 3sin x 4sin' a;.

Tan 3a: = tan (2x + x)= . ten f x TLJ^JH. i n the latter substitute for tan 2x its

1 tan 2a; tan x

value in terms of tan x.

13. Prove that sin 4# = 4 (sin x 2sin 8 ic) cosa;.

2tan x 2

14. Prove that sin x

tan a \x cot \x + tan J

2sin 4a? cos 4a? 2 tan A 2 tan

BUG. Sin x = 2sin ia? cos \x = = - p_ = _

seci secia; sec'Ja; 1+la

COSia:

produce the last form divide numerator and denominator of . - |^- bv

a *

i K r> LI i. i 1 COS iC A , 1 + COS X

15. Prove that tan \x = : - . Also cot x = - : - .

sin x sin a;

SDG. Prom (L), (56*), 2sin a x - 1 cos , and from (K), 2sin \x cos %x = sin ,

Divide the former by the latter.

FORMULA ADAPTED TO LOGARITHMIC COMPUTATION. 29

16. Find the trigonometrical functions of 18.

SOLUTION. Letting x = 18, 2x = 86, and 3 = 54, hence sin 2x = cos 3a>

But sin 2x 2sin x cos x ; and cos 3.c = 4cos* a; 3cos x ; hence 2sin x cos a? ~

4cos s x 3cos , or 2sin x = 4cos a x 3 4 4sin 3 x 3. From which 4sin' a

+ 2sin x \ Solving this quadratic, we have sin x t or sin 18 = *

neglecting the - root, since sin 18 is + . From this, cos 18 = i|/10 + 2^5.

These may be put in approximate decimal fractions.

17. Having given the functions of 18, and 15 (Ex. 8), find those

of 3 ; then of 6, 12, 24, etc.

Compare the results obtained with the values as given in the Table of Natural

Sines, etc., obtaining all the values in decimal fractions.

SECTION IIL

FORMULA FOR RENDERING CALCULABLE BY LOGARITHMS THE

ALGEBRAIC SUM OF TRIGONOMETRICAL FUNCTIONS.

58. Since multiplication, division, involution, and evolution are

the only elementary combinations of number which we can effect by

means of logarithms, if we wish to add or subtract trigonometrical

(or other) quantities, we have first to discover what products,

quotients, powers, or roots, are equivalent to the proposed sums or

differences.

59. Prop. To render sin x sin?/, and coax CQ& y calculable

by logarithms.

SOLUTION. From (55, Sen. 2) we have sin (x + y) = sin x cos y + cos z

bin y, and sin (x y) = sin x cos y cos x sin y. Adding these formulas

sin (x + y) + sin (x y} 2sin x cos y. Now putting x + y x', and x y = y'

whence x = (*' + y'}, and y = i (x f y') ; we have sin' + siny' = 2sm$(x' +V')

cos^(a;' y') ; or. dropping the accents, as the results are general,

(A') sin x + sin y = 2sin $(x + y) cos %(x y).

Again, by subtracting formula B (55, Sen. 2) from formula A, and making

he same substitutions, we have,

( B'l sin * sin y 2cos %(x + y) sin \(x y).

30 PLANE TRIGONOMETRY.

In like manner adding cos (x + y) = cos x cos y sin x sin y, and cos (x y) -

cos # cosy ^ }\nx siny, and making the same substitutions, \ve have,

(C') cos a; + cosy = 2cosi(cc + y) cos \(x - y}.

Finally subtracting formula C (55, Sen. 2) from formula D, and making

the same substitutions, we have,

(DO cos y cos x = 2sin \(x + y) sin i(x y), or

cos x cos y= 2sin \(x + y) sin %(x y).

60. COR.. 1. The sum of the sines of two angles is to their differ-

e",je, as the tangent of one-half the sum of the angles is to the tangent

of one-half their difference.

DEM. Dividing A' by B', we have,

sin x + sin y _ sin j(x + y) cos j(x y) _ sin |(a + y) ^ cos $(x y) _

sin x sin y ~~ cos i(x + y) sin %(x y) ~~ cos %(x + y) sin \(x y) ~~

tan & + y) cot - y) = tou ft, + y) x

6j? COR. 2. TAe difference of the cosines of two angles divided by

their sum is numerically equal to the product of the tangent of one-

half the sum of the two angles into the tangent of one-half their

difference.

DEM. Dividing D' by C', we have,

cos x - cos y _ sin j(x + y) sin j(x y) _ _ sin |(a? + y) sin j(x y) _

cos x + cos y ~~ cos \(x + y) cos i(x y) " cos i(x + y) cos i(x y) ~~

tan KX + y) tan $(x y). Q. E. D. (Observe the opposition in signs.)

62. _PfO&. To render tan x tany calculable ly logarithms.

sin a; sin y sins cosy cosz sin y sin ( y)

DEM. Tan x tan y = - = =

cos x cosy cos x cosy cos a cos #

B. D

EXERCISES.

Let the student deduce the following relations

sin (x + i/}

1. Cot x + cotv = . v T y/ .

sm x sin y

2 cos lx + cos iz v

Sec * - aec j, = + y) si

cosa; cosy

COKfeTUUCTION AND USE OF TRIGONOMETIUCAL TABLES. 31

4. 1 + cos x = 2cos 3 ^. (See J7.)

5. 1 cos a; =

6. sina; + sm ^ = tan 4(3 + y). (Divide A' by 0', SO.)

cos a + cos

, sm x sin y , , .

7. - -= tan4(z-y).

cos x + cos y

t sin x 4- sin y

8. - 2. _ C ot 4(z y).

cos x cos y

rins-riny^

cos x cos y

SUCTION IV.

CONSTRUCTION AND USE OF TRIGONOMETRICAL TABLES.

[NOTE. In order to read this and the subsequent sections, the student needa

a knowledge of the nature of logarithms, and the method of using common

logarithmic tables. If he is familiar with the last chapter in THE COMPLETE

SCHOOL ALGEBRA of this series, he is prepared to go on. If he has not this

knowledge, he should read the introduction preceding the table of Logarithms

before reading this section.]

63. A Table of Trigonometrical Functions is a table

containing the values of these functions corresponding to angles of

all different values. In consequence of the incommensurability of

an arc and its functions, these results can be given only approxi-

mately; yet it is possible to attain any degree of accuracy which

practical science requires.

G4i* There are two tables of trigonometrical functions in common

use, the Table of Natural Functions, and the Table of Logarithmic

Functions.

65* A Table of Natural Trigonometrical Functions

is a table in which are written the values of these functions for

angles of various values, the radius of the circle being taken as the

measuring unit, and the function being expressed in natural num-

bers extended to as many decimal places as the proposed degree of

accuracy requires.

66. A Table of Logarithmic Trigonometrical Func-

tions is the same as a table of natural functions, except that the

logarithms of the values of the functions are written instead of the

functions themselves, and to avoid tbe frequent occurrence of nes;a-

32 PLANE TRIGONOMETRY.

tive characteristics, the characteristic of each logarithm is increased

by 10. For example, sines and cosines being always less than unity,

except at the limit (33), and tangents of angles less than 45 and

cotangents of angles greater than 45 being also less than unity, the

logarithms of all such functions have negative characteristics. To

obviate the necessity of writing these with their sign, the charac-

teristic of each logarithm is increased by 10.

67. JProb. To compute a table of natural trigonometrical func-

tions for every degree and minute of the quadrant.

SOLUTION. It is evident that an arc is longer than its sine, but that this

disparity diminishes as the arc grows less. Thus, in a circle whose radius is

1 inch, the length of the sine of an arc of 1 would not differ appreciably from

the arc. Much less should we be able to distinguish between the sine of 1' and

the arc. Now, since when the radius is 1, a seinicircumference = it = 3.1415926,

and also = 180, or 180 X 60 10800', we have the length of an arc of 1' =

' = 0.0002908882 approximately. Assuming this as the sine of 1', we

obtain the cosine thus,

cos V =Vl-sin a l' =V(l + sinl')X(l-sinl') = v/1.0002908882 X .9997091118

= 0.9999999577.

Having thus obtained sufficiently accurate values of sin 1' and cos 1', we can

continue the operation as follows : from the formulae sin (x + y} + sin (x y) =

2 sin x cos y, and cos (x + y) + cos (x y} = 2 cos x cos y, we have

sin (x + y) = 2 sin x cos y sin (x y\

cos (x + y) 2 cos x cos y cos (x y).

Now letting y remain constantly equal to 1', and letting x take successively

the values 1', 2', 3', etc., we have

, ( sin 2' = 2 cos V sin 1' - sin 0' = 0.0005817764

1 K

' ( cos 2' = 2 cos 1' cos V - cos 0' = 0.9999998308

j sin 3' = 2 cos 1' sin 2' - sin 1' = 0.0008726646

1 cos 3' = 2 cos 1' cos 2' - cos V = 0.9999996193

sin 4' = 2 cos 1' sin 3' sin 2' = 0.0011635526

_

' I cos 4' = 2 cos V cos 3' - cos 2' = 0.9999993232

j sin 5' = 2 cos 1' sin 4' - sin 3' = 0.0014544407

4 ' \ cos 5' = 2 cos 1' cos 4' - cos 3' = 0.9999989425

etc., etc.

These operations present no difficulties except the labor of performing the

numerical operations.

Of course 60 operations are required for every degree, and far 30, 1800. Bui

having computed the shies and cosines for every degree and minute up to 30

CONSTRUCTION AND USE OF TRIGONOMETRICAL TABLES. 33

we can complete the work by simple subtraction of values already found. For

example, letting x = 30, tbe first formula used above becomes

sin (30 + y) = cos y - sin (30 - y\

and from cos (x + y) cos (x y} = 2 sin x sin y, we have

cos (30+ y) = cos (30 - y} - sin y

Now making y successively = 1', 2', 3', etc., these give

j sin 30' 1' = cos 1' sin 29 59'

( cos 30 V = cos 29 59' sin 1'

j sin 30 2' = cos 2' - sin 29 58'

( cos 30 2' = cos 29 58' - sin 2'

j sin 30 3' = cos 3' - sin 29 57

\ cos 30 3' = cos 29 57' - sin 3'

etc., etc.

All of these values which occur in the second members having been deter-

mined in reaching sin 30 and cos 30, those in the first members can be found

by performing the requisite subtractions.

Proceeding in this way till we reach 45, the numerical values of all sines and

cosines become known, since the sine of any angle between 45 and 90", being

the cosine of the complementary angle, will have been computed in reaching

45. And so also the cosines of angles between 45 and 90 will have been

computed as sines of the complementary angles below 45.

The sines and cosines being computed, the corresponding tangents, cotan-

gents, and, if need be, the secants, cosecants, versed-sines, and coversed-

sin x 1 cos x

sines, can be calculated from the relations tana; = , cot a = or - ,

cos x 1 tan x sin x

sec a = , coseca; = . , versa; = 1 cos a;, and covers x = 1 sin a?.

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