Half-side formula

Spherical triangle

In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles.[1]

Formulas

The half-side formulas are[2]

{\displaystyle {\begin{aligned}\tan \left({\frac {a}{2}}\right)&=R\cos(S-\alpha )\\[8pt]\tan \left({\frac {b}{2}}\right)&=R\cos(S-\beta )\\[8pt]\tan \left({\frac {c}{2}}\right)&=R\cos(S-\gamma )\end{aligned}}}

where

• a, b, c are the lengths of the sides respectively opposite angles α, β, γ,
• ${\displaystyle S={\frac {1}{2}}(\alpha +\beta +\gamma )}$ is half the sum of the angles, and
• ${\displaystyle R={\sqrt {\frac {-\cos S}{\cos(S-\alpha )\cos(S-\beta )\cos(S-\gamma )}}}.}$

The three formulas are really the same formula, with the names of the variables permuted.