Cosine Rule on a Sphere

As all (well, at least those that can remember) 15+ year old’s know, to find the length of a side of a non-right angled triangle you can’t use Pythagoras theorem and instead require the so called ‘Cosine Rule’.

Simply that says, to find, for example, the length of side c in:

You use the following formula: \[c^2 = a^2 + b^2 Р2ab\cos(\gamma).\]

But what happens if we move away from Euclidean Geometry and that our triangle is now sitting on a sphere? I.e.

Logic tells us that of course the same cosine rule can not be applied to find the side c. Instead we use an alternative formula, namely: \[\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C).\] The proof is relatively straight forward, so why don’t you have a go at it before looking below!

Proof Select AllShow

The corresponding sine rule for a sphere is very easy to find, but in case you’re stuck on that, I’ll post it in a couple of days.

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