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

First Class. 31
From (82) and (85) we get
tan sinE in (C-) (88)
tan C = sin (A - E) sin B - E)
Cor. 1.-tan a. tan b = sin E sin(C - E). (89)
tan } b. tan c = sin E sin (A - E). (90)
tan. t a = sin E sin (B - E). (91)
Cor. 2.-tan a. cot 1 b = sin (A - E) sin (B - E). (92)
tan b. cot 1 c = sin (B - -E) sin (C- E). (93)
tan ic. cot a = sin(C - E) sin (A -E). (94)
Cor. 3.tan ja: tan b: tan ic:: sin (A - E): sin(B- E): sin (C-E).
(95)
ExERacSES. —VI.
1. Prove cosC = - cos (A + B) cos - os (A - B) sin2 c (96)
N sin E
2. sin ~ a sin ~ b sin N - (97)
2.",2, si s 9 a =sin A sin B sin C'
where N = V sin Esin (A - -E) sin (B - E) sin (0- E). (98)
N is called the Norrm of the angles of the triangle. See note, ~ 28, Ex. 5.
VN2
3. Prove that cos a cos b cos c = sin s (99)
sin E sin A sin B sin C'
sin2 E
4.,, tan ltan btanc = s-. (100)
2N 2r 2N
5. sin a = sinC sin b = in -' sin C=.
sin B sin GC sin C sin A sin A sin A '
(101)
The value N = sin B sin C sin a, and the corresponding value of n, viz..
a sin b sin c sin A, have a remarkable analogy to the equation S =bc sin A
in plane trigonometry for the area of a triangle.
sin A sin B sin C N
6. Prove si= -a =.-.-. (102)
sin a sin b sin c n