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
.
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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