Law of Cosines - Law of Cosines in Non-Euclidean Geometry

Law of Cosines in Non-Euclidean Geometry

A version of the law of cosines also holds in non-Euclidean geometry. In spherical geometry, a triangle is defined by three points u, v, and w on the unit sphere, and the arcs of great circles connecting those points. If these great circles make angles A, B, and C with opposite sides a, b, c then the spherical law of cosines asserts that each of the following relationships hold:

\begin{align} \cos a &= \cos b\cos c + \sin b\sin c\cos A\\ \cos A &= -\cos B\cos C + \sin B\sin C\cos a. \end{align}

In hyperbolic geometry, a pair of equations are collectively known as the hyperbolic law of cosines. The first is

where sinh and cosh are the hyperbolic sine and cosine, and the second is

Like in Euclidean geometry, one can use the law of cosines to determine the angles A, B, C from the knowledge of the sides a, b, c. However, unlike Euclidean geometry, the reverse is also possible in each of the models of non-Euclidean geometry: the angles A, B, C determine the sides a, b, c.

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