Spherical Law of Cosines

​
vertex A
θ
1.047
vertex B
θ
1.047
ϕ
1.083
show sphere
Draw a spherical triangle
ABC
on the surface of the unit sphere with center at the origin
O
. Let the sides (arcs) opposite the vertices have lengths
a
,
b
and
c
, and let
A
be the angle at vertex
A
. The spherical law of cosines is then given by
cosa=cosbcosc+sinbsinccosA
, with two analogs obtained by permutations.

Details

Let
p
be the plane tangent to the sphere at
A
, and let
Q=p⋂OB
and
R=p⋂OC
. Then
|OQ|=1/cosc
,
|AQ|=sinc/cosc
,
|OR|=1/cosb
and
|AR|=sinb/cosb
. Express the length of
|QR|
in two ways using the usual planar law of cosines for the triangle
AQR
in the plane
p
:
|QR|=
2
sinb
cosb
+
2
sinc
cosc
-2
sinb
cosb
sinc
cosc
cosA
.
With the triangle
OQR
, the law of cosines gives
QR=
2
1
cosb
+
2
1
cosc
-2
1
cosb
1
cosc
cosa
.
The two equalities give
cosa=cosbcosc+sinbsinccosA
.

References

[1] Wikipedia. "Spherical Law of Cosines." (Feb 22, 2017) en.wikipedia.org/wiki/Spherical_law_of _cosines.

External Links

Spherical Trigonometry (Wolfram MathWorld)
Spherical Triangle Solutions

Permanent Citation

Izidor Hafner
​
​"Spherical Law of Cosines"​
​http://demonstrations.wolfram.com/SphericalLawOfCosines/​
​Wolfram Demonstrations Project​
​Published: February 23, 2017