Spherical Cosine Rule for Angles

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