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WOLFRAM NOTEBOOK
Spherical Cosine Rule for Angles
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 , , .
ABC
O=(0,0,0)
a
b
c
α
β
γ
A
B
C
Construct a supplementary spherical angle with apex and sides , , . Let , , be the vertices and , , be the plane angles at those vertices. Then:
O'
a'=O'A'
b'=O'B'
c'=O'C'
A'
B'
C'
α'
β'
γ'
a'=π-A
b'=π-B
c'=π-C
and
C'=π-c
B'=π-b
C'=π-c
These are the cosine rules for the sides of a spherical triangle:
cosa=cosbcosc+sinbsinccosα
cosb=cosccosa+sincsinacosβ
cosc=cosacosb+sinasinbcosγ
Note that and . Apply the cosine rules for the sides on the supplementary angle:
cos(π-x)=-cosx,sin(π-x)=sinx
cos(π-x)=-cosx,sin(π-x)=sinx
cos(π-α)=cos(π-β)cos(π-γ)+sin(π-β)sin(π-γ)cos(π-a)
-cosα=cosβcosγ-sinβsinγcosa
cosα=-cosβcosγ+sinβsinγcosa
In the same way,
cosβ=-cosαcosγ+sinαsinγcosb
cosγ=-cosβcosα+sinβsinαcosb
The last three identities are known as the cosine rules for the angles or the second cosine theorem for a spherical angle.