# Law of Cosines for Spherical Triangles

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## Contents

## 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

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.97$