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:

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

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.

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:

$\ds \cos a'$

$=$

$\ds \cos b' \cos c' + \sin b' \sin c' \cos A'$

$\ds \leadsto \ \ $

$\ds \map \cos {\pi - A}$

$=$

$\ds \map \cos {\pi - B} \, \map \cos {\pi - C} + \map \sin {\pi - B} \, \map \sin {\pi - C} \, \map \cos {\pi - a}$

$\ds \leadsto \ \ $

$\ds -\cos A$

$=$

$\ds \paren {-\cos B} \paren {-\cos C} + \map \sin {\pi - B} \, \map \sin {\pi - C} \, \paren {-\cos a}$

$\ds \leadsto \ \ $

$\ds -\cos A$

$=$

$\ds \paren {-\cos B} \paren {-\cos C} + \sin B \sin C \paren {-\cos a}$

$\ds \leadsto \ \ $

$\ds \cos A$

$=$

$\ds -\cos B \cos C + \sin B \sin C \cos a$

simplifying and rearranging

$\blacksquare$

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

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.97$
1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $11$. Polar formulae.
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cosine rule (law of cosines): 2.
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cosine rule (law of cosines): 2.