Sine Formula

In planar trigonometry, there is the sine formula

abc
sin A sin B sin C

In spherical trigonometry, a similar formula exists,

sin a sin b sin c

sin A sin B sin C

Obviously, for small arcs, Eq. B.15 reduces to Eq. B.14.

To prove Eq. B.15, we rewrite Eq. B.2 as

sin b sin c cos A = cos a — cos b cos c. (B.16)

Squaring, we obtain

sin2 b sin2 c cos2 A = cos2 a — 2 cos a cos b cos c + cos2 b cos2 c. (B.17)

The left-hand side can be written as  sin2 b sin2 c — sin2 b sin2 c sin2 A, (B.18)

By definition, in a spherical triangle, the sides and the vertical angles are always smaller than 180°. Therefore, in Eq. B.22, only a positive sign is admissible. Because Z is symmetric to A, B and C, we obtain

sin a sin b sin c sin A sin B sin C

B. 4 Formula C

We rewrite the cosine formula B.11 in the following form and use Eq. B.13: sin b sin c cos A = cos a — cos b cos c

= cos a — cos b (cos a cos b + sin a sin b cos C) (B.24)

= cos a sin2 b — cos b sin a sin b cos C.

Dividing both sides by sin b, one obtains formula C sin c cos A = cos a sin b — sin a cos b cos C.

Similarly,

sin a cos B = cos b sin c — sin b cos c cos A, sin a cos C = cos c sin b — sin c cos b cos A,

and so on.

Problems

B.1. Show that if one of the arcs is 180°, then no spherical triangle can be constructed.

B.2. If one of the vertex angles of a spherical triangle, for example, C, is a right angle,

show that for small arcs the cosine formula leads to the Pythagorean theorem.

B.3. Using the cosine and sine formulas, show that

cot a sin b = cot A sin C + cos b cos C, (B.28)

cot c sin a = cot C sin B + cos a cos B, (B.29)

and so on.

B.4. For a rectangular spherical triangle, where C=90°, show that

 sin a = sin c sin A, (B.30) sin b = sin c sin B, (B.31) tan a = tan c cos B, (B.32) tan b = tan c cos A, (B.33) tan a = sin b tan A, (B.34) tan b = sin a tan B. (B.35)