Plane trigonometry, by S.L. Loney.

SOLUTION OF TRIANGLES.

191

179. We now proceed to the case of the triangle which
is not given to be right angled.
The different cases to be considered are;
Case I. The three sides given;
Case II. Two sides and the included angle given;
Case III. Two sides and the angle opposite one of
them given;
Case IV. One side and two angles given;
Case V. The three angles given.
180. Case I. The three sides a, b, and c given.
Since the sides are known, the semi-perimeter s is
known and hence also the quantities s - a, s-b, and
s - C.
A B C
The half-angles,, and - are then found from the
formulae
A (s-b )( ) ), tan /(s- c)(s-a)
tanA -b (s - c
2( (s- a) 2 s(s- b)
an(s - a) (s - b)
and tan 1-=
and t = s (s - c)
Only two of the angles need be found, the third being
known since the sum of the three angles is always 180~.
The angles may also be found by using the formulae
for the sine or cosine of the semi-angles.
(Arts. 165 and 166.)
The above formulae are all suited for logarithmic
computation.