Ambiguous Case

Theorem
Let $\triangle ABC$ be a triangle.

Let the sides $a, b, c$ of $\triangle ABC$ be opposite $A, B, C$ respectively.

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

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

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

This is known as the ambiguous case.

Proof
From the Law of Sines, we have:


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

from which:
 * $\sin A = \dfrac {\sin a \sin B} {\sin b}$

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.


 * Ambiguous-Case.png

Also see

 * Ambiguous Case for Spherical Triangle