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

52 Solution of Spherical Triangles.
If the side c be very small, instead of the formulae (231),
we may determine it by means of either of the following
equationstan c = tan b - cos A = tan a - cos B, (232)
A, B having been calculated from the formulae (229), (230).
EXERCISES.-XII.
1. Given a = 120~ 10' 0", b = 150 59' 44"; find A, B, c.
2.,, a= 36 27 0, b= 43 32 31;,,,
3.,, a= 86 40 0, b= 32 40 0;,,,,
54. FOUIRT CASE.-Being given a, A, tofind c, b, B.
In this case we have
sin c = sin a ' sin, equation (108). (233)
sin b = tan a - tanA,, (110). (234)
sinB = cos A- cos a,,, (111). (235)
Since each of the sought parts is found by its sine, there will
be on the whole six solutions, the formulae (233)-(235) giving
two values for each of the sought parts c, b, B. The parts
a, A must be of the same species (see ~ 50). Also sin a
must be less than sin A (A is comprised between a and 90),
and the formula (233) gives two admissible values of c, say c,
and 180 - cl; the formula (234) gives two values of b, b, and
180 - b,, one of which goes with c,, and the other with 180 - c,;
for the three sides a, b, c are all acute, or one alone is acute; the
formula (235) gives two values of B, of which one goes with b,,
and the other with 180 - bi (b and B are of the same species).
If the sought parts are badly determined by the equations
(233)-(235), which happens when they are near 90~, the preceding formulae may be written as follows (see Notation,
~ 43):