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.


Side-Side-Angle

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 $\sphericalangle A$.


Angle-Angle-Side

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 $a$.


These are known as the .


Also see


Sources

  • 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): solution of triangles
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