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

22 Connecting Sides and Angles of a Spherical Triangle.
28. From (16) and (19), we get
tan j. = sin ( - b) in (8 - (22)
sin 8 sin (s - a)
In like manner,
tan B = sin (s - C) sin (8 - ) (23)
sin s sin (8 - b)
and tan C = sin ( - a)in ( - b) (24)
sin 8 sin (8 - c)
Cor.- tan -A tan B = sin (s- c) (25)
sin s
tan aB tan C C = 8n ( ) (26)
min 8
tan t C tan - sin ( - b) (27)
sin s
EXERCISES. -III.
1. Prove sin2i* 1 - cos2a - cos2b-cos2c + 2 cos acosb cos 28)
1. Prove sinaAX --- -- - -- (28)
sin2 b sin2 c
cos etos b cos c b sIbM^
Make use of cos A os b cos c
sin b sin c
2.,, cos c = cos (a + b) sin2 7 C + cos (a - b) cos2 O C. (29)
3., cos2 c = cos2 (a + b) sin2 + cos2(a - b) cos C. (30)
4., sin2 C = sin2 (a + b) in2 C + sin2 (a - b) cos2 C. (31)
n2
5.,, s in A sin sin C =.sin (32)
2' s in s sin a sin b sine'
where n = /sin s sin (s - a) sin (s - b) sin (s - c). (33)
The function n is so important in spherical trigonometry that it is convenient to have a definite name for it. PROF. STAUDT*, of the University
of ERLANGEN, calls 2n the sine of the trihedral angle 0- ABC (see fig., ~ 24).
NEUBERG suggests two other names-1~. The First Stacdtian of the triangle.
* CRELLE'S Journal, Band. xxiv., s. 255.