Law of Cosines for Spherical Triangles

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Theorem

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.

Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.


Then:

Sides of Triangle

$\cos a = \cos b \cos c + \sin b \sin c \cos A$


Angles of Triangle

$\cos A = - \cos B \cos B + \sin B \sin B \cos a$


Also known as

This result is also known as the Spherical Law of Cosines.


Historical Note

This result was first stated by Regiomontanus in his De Triangulis Omnimodus of 1464.


Sources