Law of Cosines for Spherical Triangles

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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.


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.