Sine of Half Side 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:
 * $\sin \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\sin B \sin C} }$

where $S = \dfrac {A + B + C} 2$.

Proof
Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.

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

From Spherical Triangle is Polar Triangle of its Polar Triangle we have that:
 * not only is $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$
 * but also $\triangle ABC$ is the polar triangle of $\triangle A'B'C'$.

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

We have:

Then:

and:

The result follows.

Also see

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


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