The spherical law of sines states that

On the sphere $\mathbb{S}^2$ we consider a triangle, i.e. three points connected by geodesics. We denote the side lengths with $a, b, c < \pi$ and the angles opposite to these sides with $\alpha, \beta, \gamma \in [0, 2\pi]$.

Then: $$ \frac{\sin \alpha}{\sin a} = \frac{\sin \beta}{\sin b} = \frac{\sin \gamma}{\sin c}.$$

As part of my studies, I have been given the task of proving the above theorem. However, I need to somehow apply Clairaut's theorem, that is:

Let $f$ be a rotation surface, i.e. $f(t, \varphi) := \left( r(t)\cos\varphi, r(t) \sin \varphi, h(t)\right)$ and let $\gamma(s) := \left( t(s), \varphi(s)\right)$ be a regular curve and $c := f \circ \gamma$. Futher, we denote by $\theta(s)$ the angle between the curve $c(s)$ and the line of latitude through $c(s)$. That means: $$ \cos \theta(s) = \frac{\langle c'(s), \partial_2f(\gamma(s)) \rangle}{\lvert c'(s) \rvert \lvert \partial_2f(\gamma(s)) \rvert}.$$

Then: $$ s \mapsto r\left(t(s)\right) \cos \theta(s) \; \text{is constant}.$$

What I did: I proved Clairaut's theorem.

Remark: I know that many answers are already given here – however I think I need a little help to apply Clairaut's theorem.

IDEA: Maybe it could be benificial to put one corner of the triangle onto the north pole.


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