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

Oblique-angled Triangles. 59
If we compare (259) and (259'), we see that A'- B'= A - B.
Hlence, from the two formulae (a) we have À' + B' = A + B;
therefore A'= A, B' = B. Lastly, from (258) and (258') we
infer that c' = c. Hence we have the following Rule:-If each
of the two values of B which are got from (257) be such as that
(A - B) and (a - b) have like signs, there are two solutions. If
only one of them satisfies this condition, there is only one triangle
that satisfies the problem. The problem is impossible when neither
of the values of B make (A - B) and (a - b) of the same sign.
Instead of the formulae (258), (259), we may use the following:
tan c = tan (a + b) + 1 cos 2(A + B) - I cos ] (A - B).
(260)
Titan ' C = 1 tan (A. + B) + L cos (a - b) - l cos (a + b).
(261)
64. From REIDT'S Analogies (~ 44) we get the following
equations:1 tan (45~ - d"') = L { tan (s' + s") + I tan (s' - s")
+ l tan (d' + d") - tan (d' - d")}. (262)
I tan (45~ - s"') = 1- { tan (s' + s") - 1 tan (s' - s")
+ l tan (d' + d") + tan (d'- d")}. (263)
These formulae determine C and c when the angle B is acute.
They possess the advantage of requiring only four logarithms
instead of six, which are necessary if we calculate by the equations (258), (259). For the second triangle answering the given
conditions, or for B obtuse, the formulae areI tan s"' - {I tan (s' + s") + 1 tan (s' - s") + I tan (d' - d")
- tan (d' + d")}. (264)
I tan d"' + ( {I tan (s' - s") + 1 tan (d' - d") + 1 tan (d' + d")
- Titan (8' + 8")}. (265)