Half Angle Formulas for Spherical Triangles
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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.
Cosine of Half Angle for Spherical Triangles
- $\cos \dfrac A 2 = \sqrt {\dfrac {\sin s \, \map \sin {s - a} } {\sin b \sin c} }$
where $s = \dfrac {a + b + c} 2$.
Sine of Half Angle for Spherical Triangles
- $\sin \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin b \sin c} }$
where $s = \dfrac {a + b + c} 2$.
Tangent of Half Angle for Spherical Triangles
- $\tan \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin s \, \map \sin {s - a} } }$
where $s = \dfrac {a + b + c} 2$.