Sine of Half Angle for Spherical Triangles

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.

Then:
 * $\sin \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin b \sin c} }$

where $s = \dfrac {a + b + c} 2$.

Proof
The result follows.

Also see

 * The other Half Angle Formulas for Spherical Triangles:
 * Cosine of Half Angle for Spherical Triangles
 * Tangent of Half Angle for Spherical Triangles


 * Half Side Formulas for Spherical Triangles:
 * Sine of Half Side for Spherical Triangles
 * Cosine of Half Side for Spherical Triangles
 * Tangent of Half Side for Spherical Triangles