T

(18) 2 cos=2+/2.

8

W2 (19) 2 cos 11° 15'=12+12+12.

sin A .sin 2A + sin A. sin 4A + sin 2A.sin 7A (20)

=tan 5 A. sin A. cos 2 A +sin 2A.cos 5 A + sin A.cos 8A

sin 0 + sin (0+$) + sin (@+20) (21)

=tan (0+0).
cos 8+cos (0+$)+cos (0+20)
(22) 2 coss A - 2 sin: A =cos 2A (1 + cos2 2A).
(23) (3 sin A - 4 sin? A)2+(4 cos3 A - 3 cos A)2=1.

sin 2a. cos a
(1 + cos 2a) (1 + cos a

cot (n − 2)a.cot na tl (25) 2

=cota - tan a. cot (n − 2) a-cot na (26) If tan a=7 and tan B= ii, prove tan (2a +B)=.

A

А (27) Prove that tan and cot

2

are the roots of the equation % - 2x, cosec A+1=0. 6

2 cos B (28) If tan B=

b a+b+cos 2B

=tan

a

(24 )

=

ab

-, prove that

+ sin

* 168. The following examples are symmetrical, and each involve more than two angles :

Example 1. Prove that sin (a +B+3)=sin a. cos ß.cos y+sin ß. cos y. cos a

cos a.cos B – sin a.sin ß. sin y. sin (a +B+y)=sin (a+b). cos y +cos (a +ß) sin y =sin a.cos ß.cos y + cos a.

sin ß.cos y + cos a.cos ß.sin y-sin a. sin ß.sin y =sin a.cos ß. cos y+sin ß. cos y. cos a + sin y. cos a. cos ß - sin a.

sin ß. sin y.

Q.E.D.

.

gta

Example 2. Prove that
sin a + sin ß+sin y-sin (a +B+y)

B+y
=4.sin sin
2

2

2a+B+y Now sin a - sin (a +B+y)=

= -2 cos.

sin 2

And sin ß+sin y=2 sin

2

2 .. sin a + sin ß+sin y-sin (a +B+y)

B+y

2

B+y

COS

BY, [Art. 158]

2a +B+ y sin 2 cos

COS

2

=2 sin B+y
=2 sin B+Y/cos B-7

B
COS

2

2

2
=2 sin.
B+y

2 sin
gta

sin

a+B 2

2

[Art. 158]

=4 sin $#%. sin +

Prove the following statements :
(1) cos (a+B+y)=cos a.cos ß.cos y -cos a. sin ß. sin y

- cos ß. sin y. sin a- cos y. sin a. (2) sin (a +ß-x)=sin a.cos ß.cos y +sin ß. cosy.cos a

- sin y,cos a. cos ß+sin a sin ß sin y. (3) cos (a - B+x)=cos a.cos ß. cos y + cos a. sin ß.sin y

- cos ß. sin a. sin y+cos y sin ß. sin a. (4) sin a+sin B – sin g - sin (a+B-y)

a+ß
=4 sin

9
2
2

2 (5) sin (a - B-7) – sin a + sin ß+sing

=4 sin
a-ß
sin

- y
2
2

2 (6) sin 2a + sin 28+sin 2y-sin 2(a +B+y)

=4 sin (B+y). sin (y + a). sin (a +B).

Y +2

COS

COS

2

+

(7) sin (8-y)+sin(y-a) + sin(a - b)

)
(B-»), sin (75a), sin (45B)=0.
+4 sin
y-

.
2
2

2 (8) sin (B+y-a)+sin(y+a-B) + sin (a +B-)

- sin (a + B+y)=4 sin a.sin ß.sin y. (9) sin (a +B+y)+sin (B+y-a) +sin (y+a-B)

- sin (a +ß - y)=4 cos a.cos ß.sin y. (10) cos x+cos y + cos z+cos (x+y+z)

2+36 =4 cos

2 2 (11) cos 2x + cos 2y + cos 22 + cos 2 (x+y+z)

=4 cos (y + 2). cos (+x). cos (x+y). (12) cos(y+2– X) + cos (2+x-y)+cos (x +y - 2)

+cos (x+y+z)=4 cos x.cosy.cos 2. (13) cos2x+coso y + cosa 2+ cos2 (6+y+z)

=2 {1+cos (y+z). cos (2+x). cos (x+y)}. (14) sina x +sin? y +sina z+sin(x+y+z)

=2 {1 - cos (y +2). cos (2+x). cos (x+y)}. (15) cos* x + cos2 y + cosa 2 + cos2 (0+y-2)

=2 {1+cos (x - 2). cos (y-2). cos (26+y)}. (16) cosa .sin (ß-)+ cos ß.sin(y-a) + cosy.sin (a – B)=0. (17) sin a.sin (B - y) +sin ß.sin (y-a) +sin y. sin (a-B)=0. (18) cos (a +B). cos (a -B) + sin (B+ y) sin (B-y)

- cos (a +y). cos (a-y)=0. (19) cos (8-a). sin (ß-x)+cos (8-B). sin (y-a)

- cos (8-y). sin (B-a)=0.
x+ 8-0

0+ - x (20) 8 cos 2

2 =cos 20 + cos 20+cos 2x + 4 cos 0.cos $. cos x +1.

.

.

.

0+$+x.co $+x=0.

COS

COS

COS

2

CHAPTER XIII.* *

ON ANGLES UNLIMITED IN MAGNITUDE.

II.

169. The words of the proofs (on pages 118, 119) of the ‘A, B' formulæ apply to angles of any magnitude. The figures will be different for angles of different mag. nitude.

170. The figure for the 'A – B' formulæ on page 119 is the same for all cases in which A and B are each less than 90°.

The figure given below is for the proof of the 'A + B' formulæ, when, A and B being each less than 90°, their sum is greater than 90°.

F

Р

The words of the proof are precisely those of page 118. .
We
may

notice however that
ОМ Г

- M07 MK + OK OK MK cos (A + B)

OP
OP OP

ОРОР ? and the rest follows as on page 118.

171. Thus we have proved that the 'A, B' formula are true provided A and B each lie between 0° and 90°.

The student can prove them for any other values by drawing the proper figure.

The 'A, B' formulæ are therefore true for any values whatever of the angles A and B.

172. By the aid of the ‘A, B’formulæ we can prove the formula of Art. 140.

=

Example. Prove that sin (90° + A)=cos A.
sin (90°+A)=sin 90° cos A +cos 90° sin A,

=1xcos A +0 x sin A,
=cos A. Q. E. D.

EXAMPLES. XLIV.

Draw the figures for the first four of the following examples.

(1) For the (A+B) formulæ, when A is greater than 90° and (A + B) less than 180°.

(2) For the (A - B) formulæ, when A and B each lie between 90° and 180°.

(3) For the (A + B) formulæ, when A lies between 90o and 180°, and (A + B) lies between 180° and 270°.

(4) For the (A - B) formulæ, when A lies between 180° and 270°, and (A - B) lies between 180° and A.

Deduce the six following formulæ from the 'A, B' formulæ. (5) cos (90° + A)= - sin A. (6) sin (90° – A)=cos A. (7) cos (900 – A)=sin A. (8) sin (180° – A)=sin A. (9) cos (180° – A)= -cos A. (10) sin (1809+ A)= - sin A.

(11) Assuming that the formula sin (A + B)=sin A.cos B +cos A. sin B is true for all values of A and B, deduce the rest of the ‘A, B' formulæ by the aid of the results on p. 107.