Spherical Law of Sines

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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 , and be the angles at the vertices , and . The spherical law of sines states that .

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



Let be the orthogonal projection of onto the plane . Let and be the orthogonal projections of onto the lines and , respectively. Then , , , and , so .


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

[2] Wikipedia. "Spherical Trigonometry." (Feb 22, 2017) en.wikipedia.org/wiki/Spherical_trigonometry.

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