Ambiguous Case for Spherical Triangle
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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).
Proof
From the Spherical Law of Sines, we have:
- $\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$
from which:
- $\sin A = \dfrac {\sin a \sin A} {\sin b}$
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.
$\blacksquare$
Also see
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $6$. The sine-formula.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ambiguous case
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ambiguous case