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

Oblique-angled riangles. 63
67. Given two angles A, B, and the adjacent side c, to finJ
a, b, C.
From Napier's Analogies we get
I tan 1 (a + b) = 1 tan- c + 1 cos (L(A - B) - i cos 2- (A + B).
(270)
tan 1 (a - b) = I tan 1 c + 1 sin (A B - B) - 1 sin (A. + B).
(271)
Hence a, b are known, and C can be found from (268) or (269).
Or thus:-Let fall the perpendicular BE (see last fig.); then
denoting the angle ABD by <, the angle DBC will be B -.
Then from the triangles ABD, CBD, we get
cot b = tan A cos c, tan a = cos s tan C/cos (B - ).
The first of these formulae determines <, and the second a.
Similarly b may be found.
Again, from the same triangles, we have
sin 4: sin (B - ):: cos: cos C.
Hence C is found.
68. The following simple and elementary methods of solving
the various cases of oblique-angled triangles, by dividing each
into two right-angled triangles, are due to CAUCHY.
_ c
\ 1/ D
Fig. 21.
Let 0 be the centre of the sphere, AB C the spherical triangle.