Half Side Formulas 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.

Cosine of Half Side for Spherical Triangles

$\cos \dfrac a 2 = \sqrt {\dfrac {\map \cos {S - B} \, \map \cos {S - C} } {\sin B \sin C} }$

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

Sine of Half Side for Spherical Triangles

$\sin \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\sin B \sin C} }$

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

Tangent of Half Side for Spherical Triangles

$\tan \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\map \cos {S - B} \, \map \cos {S - C} } }$

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

Also known as

Some sources render Half Side Formulas for Spherical Triangles with a hyphen: Half-Side Formulas for Spherical Triangles.

Some use the term Half-Side Formulae for Spherical Triangles.

Also see

• Half Angle Formulas for Spherical Triangles