# Law of Sines for Spherical Triangles

## 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:

$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$

## Also known as

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

## Historical Note

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

## Sources

... where he misattributes it to Georg Joachim Rhaeticus