Spherical Right Triangles

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Draw a spherical triangle on the surface of a unit sphere centered at . Denote the arcs opposite the corresponding vertices as , , . Let , , be the angles at the vertices , , . Suppose , so that these become spherical right triangles. The following relations are then valid:

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


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Details

Let be the orthogonal projection of onto the edge . Let be the orthogonal projection of onto . Then , , but also . Thus .

, , but also . Thus .

References

[1] Wikipedia. "Spherical Law of Cosines." (Mar 21, 2017) en.wikipedia.org/wiki/Spherical_law_of _cosines.

[2] Wikipedia. "Spherical Trigonometry." (Mar 21, 2017) en.wikipedia.org/wiki/Spherical_trigonometry.



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