Four-Parts Formula

Theorem
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.

Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.

We have:
 * $\cos a \cos C = \sin a \cot b - \sin C \cot B$

That is:
 * $\map \cos {\text {inner side} } \cdot \map \cos {\text {inner angle} } = \map \sin {\text {inner side} } \cdot \map \cot {\text {other side} } - \map \sin {\text {inner angle} } \cdot \map \cot {\text {other angle} }$

This is known as the four-parts formula, as it defines the relationship between each of four consecutive parts of $\triangle ABC$.

Proof

 * SphericalTriangle-FourParts.png

Also see

 * Definition:Inner Side of Spherical Triangle
 * Definition:Inner Angle of Spherical Triangle