Is there a proof of the spherical law of sines?

Question

I know that the spherical law of sines is $\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}$ but how does one prove this rule?

Answer 1

Choose a coordinate system so that the three vertices of the spherical triangle is located at $$(1,0,0),\quad (\cos a,\sin a,0)\quad\text{ and }\quad(\cos b, \sin b\cos C, \sin b\sin C)$$ The volume of the tetrahedron formed from these 3 vertices and the origin is $\frac16 \sin a\sin b\sin C$. Since this volume is invariant under cyclic relabeling of the sides and angles, we have $$\sin a\sin b\sin C = \sin b\sin c\sin A = \sin c \sin a \sin B \implies\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}$$