A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

Oblique-angled Triangles. 61
65. Given two angles A, B, and the side a opposite to one of
them, to solve the triangle.
The solution may be inferred at once from the reciprocal
case, ~~ 63, 64. In fact, the same equations solve both cases.
Third Pair of Cases.
66. Being given the sides a, b, and the contained angle C, to
find, B, c.
Napier's analogies give
tan I (A + B) = 1 cot} C+ 1 cos 1 (a - b) - I cos 1 (a + b).
(268)
I tan } (A - B) = I cot - C + I sin - (a - b) - 1 sin 1 (a + b).
(269)
These equations give 1 (A + B) and 1 (A - B); and therefore
A and B, and then c, can be found from equation (47), or from
(159) or (160).
EXAMPLE.-Given
a = 113 2 56", b = 82~ 39' 28", C= 1380 50'14"; find, B, c.
Type of the Calculation.
3(a-b) = 15~ 11' 14" 1 sin (a - b) = 1-4184891
-(a+b)=97 51 12 sina-(a + b) = 1-9959075
6 C = 69 25 7 1 cos 1 (a - b) = 1-9845438
{- cos (a+ b) = 1-1355722
cot 3 C = 1-5746163
Hence l{-tan- (A+B)} = 4235869.. I(A + B) = 110~ 39' 35",
and l tan 1 (A - B) = 2-9971969;
- (A - B) = 5~ 40' 27".
Hence A = 116~ 20' 2", B= 104~ 59' 8".