# Spherical Law of Cosines/Angles

## 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:

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

## Proof

Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.

Let the sides $a', b', c'$ of $\triangle A'B'C'$ be opposite $A', B', C'$ respectively.

From Spherical Triangle is Polar Triangle of its Polar Triangle we have that:

not only is $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$
but also $\triangle ABC$ is the polar triangle of $\triangle A'B'C'$.

We have:

 $\ds \cos a'$ $=$ $\ds \cos b' \cos c' + \sin b' \sin c' \cos A'$ Spherical Law of Cosines $\ds \leadsto \ \$ $\ds \map \cos {\pi - A}$ $=$ $\ds \map \cos {\pi - B} \, \map \cos {\pi - C} + \map \sin {\pi - B} \, \map \sin {\pi - C} \, \map \cos {\pi - a}$ Side of Spherical Triangle is Supplement of Angle of Polar Triangle $\ds \leadsto \ \$ $\ds -\cos A$ $=$ $\ds \paren {-\cos B} \paren {-\cos C} + \map \sin {\pi - B} \, \map \sin {\pi - C} \, \paren {-\cos a}$ Cosine of Supplementary Angle $\ds \leadsto \ \$ $\ds -\cos A$ $=$ $\ds \paren {-\cos B} \paren {-\cos C} + \sin B \sin C \paren {-\cos a}$ Sine of Supplementary Angle $\ds \leadsto \ \$ $\ds \cos A$ $=$ $\ds -\cos B \cos C + \sin B \sin C \cos a$ simplifying and rearranging

$\blacksquare$

## Historical Note

The Spherical Law of Cosines was first stated by Regiomontanus in his De Triangulis Omnimodus of $1464$.