Proofs for the Spherical Laws of sines and Cosines

Question

I am looking for "classical" proofs for the spherical laws of sines and cosines. A proof that relies only on knowledge that was common to the ancient greek geometers, not containing analytical geometry and vectors and such. Thanks!

Answer 1

Let $A$, $B$, $C$ the vertices of a spherical triangle on a sphere of unit radius and center $O$ (see diagram below). Spherical angle $\angle ACB=\gamma$ is by definition angle $\angle ECF$ formed by tangent lines at $C$.

In right triangles $OCE$ and $OCF$ we have then $EC=\tan b$ and $FC=\tan a$, so by the cosine rule in triangle $CEF$:

$$ EF^2=\tan^2 a+\tan^2 b-2\tan a\tan b\cos\gamma. $$ On the other hand we also have $OE=\sec b$ and $OF=\sec a$, so by the cosine rule in triangle $OEF$: $$ EF^2=\sec^2 a+\sec^2 b-2\sec a\sec b\cos c. $$ Equating the right hand sides of both formulas leads, after some simplifications, to: $$ \cos c=\cos a\cos b+\sin a \sin b\cos\gamma, $$ which is the spherical cosine rule.

This proof works as long as $a$ and $b$ are acute angles, but I think it can also be extended, with some modifications, to the other cases.