Spherical Law of Cosines

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Draw a spherical triangle on the surface of the unit sphere with center at the origin . Let the sides (arcs) opposite the vertices have lengths , and , and let be the angle at vertex . The spherical law of cosines is then given by , with two analogs obtained by permutations.

Contributed by: Izidor Hafner (February 2017)
Open content licensed under CC BY-NC-SA


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Details

Let be the plane tangent to the sphere at , and let and . Then , , and . Express the length of in two ways using the usual planar law of cosines for the triangle in the plane :

.

With the triangle , the law of cosines gives

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The two equalities give

.

Reference

[1] Wikipedia. "Spherical Law of Cosines." (Feb 22, 2017) en.wikipedia.org/wiki/Spherical_law_of _cosines.



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