Ambiguous Case for Spherical Triangle

Theorem

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.

Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.

Let the sides $a$ and $b$ be known.

Let the angle $\sphericalangle B$ also be known.

Then it may not be possible to know the value of $\sphericalangle A$.

This is known as the ambiguous case (for the spherical triangle).

$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$

from which:

$\sin A = \dfrac {\sin a \sin A} {\sin b}$

We find that $0 < \sin A \le 1$.

We have that:

$\sin A = \map \sin {\pi - A}$

and so unless $\sin A = 1$ and so $A = \dfrac \pi 2$, it is not possible to tell which of $A$ or $\pi - A$ provides the correct solution.

$\blacksquare$