Astronomy Principles and Practice Fourth Edition.pdf

The small spherical triangle 55
7.4.5 Proof of the four-parts formula
Let the four consecutive parts be B, a, C and b. Thensidea, between the angles, is the ‘inner side’, b
being the ‘other’; angle C, between sides a and b, is the ‘inner angle’ with B the ‘other’.
We want to prove that
cos a cos C = sin a cot b − sin C cot B.
From the cosine formulas (7.5) and (7.6), we have
cos b = cos c cos a + sin c sin a cos B (7.5)
cos c = cos a cos b + sin a sin b cos C. (7.6)
Substituting the right-hand side of equation (7.6) for cos c in equation (7.5), we obtain
cos b(1 − cos 2 a) = cos a sin a sin b cos C + sin c sin a cos B.
Substituting sin 2 a for (1 − cos 2 a) and dividing throughout by sin a sin b,weget
cot b sin a = cos a cos C + sin c cos B. (7.14)
sin b
Using the sine formula (7.7) it is seen that
Hence, equation (7.14) becomes
sin c
sin b = sin C
sin B .
cos a cos C = sin a cot b − sin C cot B. (7.9)
7.5 Other formulas of spherical trigonometry
1. If s = (a + b + c)/2,
sin A ( ) sin(s − b) sin(s − c) 1/2
2 = .
sin b sin c
Obviously there are two variations of this, for sin B/2andsinC/2.
2. By writing 180 − a for A, 180 − b for B, 180 − c for C, 180 − A for a, etc, in the four main
formulas, other useful formulas can be obtained. For example, using the cosine formula (7.4), we
obtain
cos A =−cos B cos C + sin B sin C cos a.
Proofs of these formulas will not be given here.
7.6 The small spherical triangle
When a spherical triangle becomes smaller and smaller on a sphere of fixed radius, its angles remain
finite but its sides tend to zero length. The triangle, in fact, approximates more and more to a plane
triangle and we would expect that the formulas of spherical trigonometry would degenerate into the
well-known formulas of plane trigonometry. This indeed is the case.