Ambiguous Case for Spherical Triangle/Angle-Angle-Side

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 angles $\sphericalangle A$ and $\sphericalangle B$ be known.

Let the side $b$ also be known.

Then it may not be possible to know the value of $a$.

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

from which:

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

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.

$\blacksquare$

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ambiguous case
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): solution of triangles
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ambiguous case
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): solution of triangles