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

The Right-angled Triangle. 49
SECTION I.-THEr RGIHT-ANGLED TBIANGLE.
50. The solution of right-angled spherical triangles presents
six distinct cases, which correspond to and are solved by the
six equations (108)-(113) of ~ 35. For their discussions the
following remarks are useful:-1~. The three sides of a spherical
triangle (omitting triangles birectangular or trirectangular), are
either ail acute, or else one is acute and the other two obtuse.
This follows from the equation cos c = cos a cos b. 2~. Either
of the sides containing the right angle is of the same species as the
opposite angle. This can be inferred from the equation
cos A = cos a sin B.
It will be a useful exercise for the student to prove these
propositions geometrically.
51. FIRST CASE.-Being given c and a, to calculate b, A, B.
The required parts are given by the formulae
COS ~
cos b= -, equation (113). (218)
cos a
sin a
sin in c' (108). (219)
tan a
cosB= tan e' (109). (220)
The formula (219) gives two supplementary values of sin A,
but the ambiguity is removed by considering that A must be of
the same species as a.
From the equations (218)-(220) we get, by obvious transformations, the three following:tan i b = + V/tan - (c + a) tan - (c - a). (221)
tan (45 + )A) = I /t-n (C- ) (222)
au-lanec - a( )
tan B=+ in (c - a) (223)
sin (c + a)
E