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

Formulae Relative to E. 87
but cos A'B' =cos a cos b + sin a sin 2 b cos C,
from the triangle A'B' C.
Hence
cos - a cos 2 b + sin 2 a sin 1 b cos (
~~cos.E= _____cos_ C *os
co-E —. (356)
COS 2 C
And substituting for cos C its value from equation (15), and
reducing, we get
E cos2-a + cos2- b + cos21 C - 1
2 cos - a cos - b cos (3
Or thus: cos b cos AA' cos -
acc~ b
os cos -' + cos 2 cos 'B' cos -.
2 2
Hence, eliminating cos AA', we get
a
2 cos b + cos c
C,osi' 2 1 + cos a + cos b + cos c
a b a b
4 cos - cos - 4 cos - cos
Hence from (355), (358)
1 + cos a + cos b + cos c
cos E = a b (359)
4 cos - cos cos2 2 2
Cor. 1.-From (351) and (354) we get
cota cot b
cot= co + cot C; (360)
sin C
and by interchange of letters we get
cot E cot cot -. (361)
sin A
and
cot - c cot - a
cot E + cot B. (362)
sin B
Cor. 2.-If the area of a spherical triangle and one of its
angles be given, the product of the semitangents of the containing sides is given.