Plane trigonometry, by S.L. Loney.

SIDES AND ANGLES OF A TRIANGLE. 175,164. In any triangle to find the cosine of an angle in
te'rms, of the sides.
A/ A A
b~ b
B aBa D[D B 0
a D a. a
Let ABC be the triangle and let the perpendicular
from A on BC meet it, produced if necessary, in the
point D.
First, let the angle C be acute, as in the left-hand
figure.
By Euc. II. 13, we have
AB2= BC2 + CA2- 2BC. CD............(i).
CD
But -= cos C, so that CD = b cos c.
Hence (i) becomes
c2 = a2 + b2 - 2a. b cos C,
i.e. 2ab cos C= a2+ b2 - c,
a2 + b2 - c2
i.e. cos C= 2 b
2ab
Secondly, let the angle C be obtuse, as in the righthand figure.
By Eue. II. 12, we have
AB2 = BC2 + CA + 2B. CD.........(ii).
CD
But = ACD = cos AC= - Co(1800- = - cos C
(Art. 72)
so that CD =-b cos C.