Plane trigonometry, by S.L. Loney.

176

TRIGONOMETRY.

Hence (ii) becomes
c2= a2 + b2 + 2a (- b cos C) = a2 + b2 - 2ab cos C,
so that, as in the first case, we have
a2 + b2 _ c2
cosC=
2ab
In a similar manner it may be shewn that
b2 + 2 - a2
cos A =-2b
2be'

and

c2 + a2 - b2
cos B = -2
2ca

If one of the angles, C, be a right angle, the above
formula would give c2=a2+b2, so that cos C= 0. This
is correct, since C is a right angle.
The above formula is therefore true for all values of C.
Ex. If a=15, b=36, and c=39,
tn 362 + 392 - 152 32 (122 + 132 - 52) 288 12
then eS 2 x 36 x 39 2x32x12 x 13 24 x 13 13
165. To find the sines of half the angles in terms of
the sides.
In any triangle we have, by Art. 164,
b2 + C _ a2
cos A = —
2be
By Art. 109, we have
cos A = 1 - 2 sin
2
TJ 0 * A &b2 C2+c-a
Hence 2 sin = -cos A = 1 -
2 2bc
2bc- b2 - c2 + a2 a2-(b2 + c2- 2bc) a2- (b- )2
2be 2b 2be
[a + (b - c)] [a - (b - c)] (a + b - c) (a- b + c)
2bc 2bc..( )