Spherical Law of Cosines (cosine rule for angles)

Description

In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.

Spherical triangle solved by the law of cosines.
A variation on the law of cosines, the second spherical law of cosines, (also called the cosine rule for angles) is described.
Given a unit sphere, a “spherical triangle” on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere (shown at right). If the lengths of these three sides are a (from u to v), b (from u to w), and c (from v to w), and the angle of the corner opposite c is C, then the (second ) spherical law of cosines is as shown here.

A and B are the angles of the corners opposite to sides a and b, respectively. It can be obtained from consideration of a spherical triangle dual to the given one.