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

38 Connecting Sides and Angles of a Spherical Triangle.
5~. sin2p = sin a' sin b'. 6~. sinp sin c = sin a sin b. (142)
7~. tan a tan b = tan c sin p. 8~. tana + tan2 b tan2ccos2p. (144)
9~. cotA: cotB:: sina': sin b'. (145)
35. If MA', MB', MC' be the perpendiculars let fall from a point M on
the sides of the triangle ABC; then
cos AB'. cos BC'. cos CA' = cos A'B. cos B'C. cos C'A. (STEINER)
(146)
36. If the triangles ABC, a8y be such that perpendiculars let fall from
A, B, C on the sides of a, j, y be concurrent, the perpendiculars from
a, B, y on the sides of ABC are concurrent.
37-40. If AD be the altitude of the triangle ABC, prove1~. cos BD: cos CD:: cos BA: cos CA. (147)
2~. sin BD: sin CD:: cot B: cot C. (148)
3~. tan BD: tan CD tanBAD: tan CAD. (149)
4~. cos BAD: cos CAD: tanBA: tan CA. (150)
41. If the base BC be fixed, and the ratio of the cosines of the sides constant, the locus of A is a great circle perpendicular to BC.
42. If the angle A be fixed, and the ratio tan b: tan c constant, the side
BC passes through a fixed point.
43. If the base BC be fixed, and the ratio tan B: tan C constant, the
locus of A is a great circle.
44. If the angle A be fixed, and the ratio cos B: cos C constant, the side
BC passes through a fixed point.
39. Quadrantal Triangles.
The triangle supplemental to a right-angled triangle has
one side a quadrant, and is called a quadrantal triangle. The
formulae pertaining to such triangles are obtained from the
equations (108)-(113) by the substitutions of the Scholium
(Art. 33). They are as follows, c being the quadrantal side:sin a = sin 4 - sin C. (151)
cos a = - tan B + tan C. (152)
tan a = tan A4 sin B. (153)
sin a = cos b cos B. (154)
cos C = - cot a. cot b. (155)
cos C= - cos A. cos B. (156)