Spherical Cosine Rule for Angles

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Let be a spherical triangle on the surface of a unit sphere centered at . Let the arcs opposite the corresponding vertices be , , . Let , , be the angles at the vertices , , .

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Construct a supplementary spherical angle with apex and sides , , . Let , , be the vertices and , , be the plane angles at those vertices. Then:

,

,

,

and

,

,

.

These are the cosine rules for the sides of a spherical triangle:

,

,

.

Note that and . Apply the cosine rules for the sides on the supplementary angle:

,

,

.

In the same way,

,

.

The last three identities are known as the cosine rules for the angles or the second cosine theorem for a spherical angle.

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


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Reference

[1] Wikipedia. "Spherical Law of Cosines." (Apr 5, 2017) en.wikipedia.org/wiki/Spherical_law_of _cosines.



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