Sine Rule on a Sphere

A couple of days ago I wrote a post on what, and how to find, the cosine rule on a sphere. In this post I’ll show you the sine rule on a sphere.

Consider the following triangle on a sphere:

The cosine rule (on a sphere) is: \[\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C).\]

The sine rule is then, where \(A\) is the angle at \(\textbf{w}\) (opposite side \(a\)) and \(B\) is the angle at \(\textbf{v}\) (opposite side \(b\)): \[\frac{\sin(A)}{\sin(a)}=\frac{\sin(B)}{\sin(b)}=\frac{\sin(C)}{\sin(c)}.\]

The proof is a simply a rearranging exercise from the cosine rule, with the formula \(\cos^2(C)+\sin^2(C)=1\). Once again I’ll provide the proof below, but I really recommend having a go yourself!

Proof Select AllShow
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