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

Pirst Class. 33
From the equation
sin A sin C sin a
sin a sin (~ 29), we get sin A = s. (108)
Sm a smc ' ~sin s
From the equation
cot c sin b = cot C sin A + cos A cos B (~ 30),
tan b
we get cos =t. (109)
tan c
From the equation
cot a sin b = cot A sin C + cos C cos b (~ 30),
tan a
we get tanA -. (110)
From the equation
cos =- cos Ccos + sin C sin cosb (~ 31),
we get sin A = b (111)
cos b'
From the equation
cos C= - cos A cos B + sin A sin B cos (~ 31),
we get cos c = cot A cot B. (112)
From the equation
cos c = cos a cos b + sin a sin b cos C (~ 24),
we get cos c = cos a cos b. (113)
36. The formulae (108)-(113) may be proved geometrically
as follows:-Let ABC be the triangle, C the right angle, 0
the centre of the sphere. From any point D in OA erect D)F
at right angles to OA, meeting OC in F, and draw FE at right
angles to OC. Join DE. Then FE is perpendicular to FD,
because the plane BO C is perpendicular toA O C. Hence
DE2 = D.F2 + FE2= O.F2- OD + OE2 - OF2= OE2 - OD;
therefore the angle ODE is right.
D