Plane trigonometry, by S.L. Loney.

182 TRIGONOMETRY.
172. Ex. From the formulae of Art. 164 deduce those of Art. 170
and vice versa.
The first and third formulae of Art. 164 give
a2 + b2-c2 c2 +a2 b2
b cos C+c cos B= - +
2a 2a
2a2
- = -a,
2a
so that a=b cos C+ c cos B.
Similarly, the other formulae of Art. 170 may be obtained.
Again, the three formulae of Art. 170 give
a= b cos C-+ccos B,
b=c cos A - a cos C,
and c=a cos B + b cos A.
Multiplying these in succession by a, b, and - c we have, by addition,
a2 + b2 - c2=a (b cos C +c cos B) + b (c cos A - a cos C) - c (a cos B + b cos A)
=2ab cos C.
a2+ b2- c2.~. cos C=-.
2ab
Similarly, the other formulae of Art. 162 may be found.
173. The student will often meet with identities, which he is required
to prove, which involve both the sides and the angles of a triangle.
It is, in general, desirable in the identity to substitute for the sides in
terms of the angles, or to substitute for the ratios of the angles in terms
of the sides.
B - C A
Ex. 1. Prove that a cos -2= (b + c) sin 2.
By Art. 163 we have
B+C B-C
2 sin cos
b + c sin B+ sin C 2sin + 2
a sin A A A
2 sin - cos -
2 2
A B-C B-C
cos - cos cos 2 -
COS - COS
2 2 2. A A A
sin cos - sin
A B-C. (b+c)sin =a cosB -
2 2