Half-side formula
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. For a triangle △ A B C {\displaystyle \triangle ABC} on a sphere, the half-side formula is tan 1 2 a = − cos ( S ) cos ( S − A ) cos ( S − B ) cos ( S − C ) {\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}a&={\sqrt {\frac {-\cos(S)\,\cos(S-A)}{\cos(S-B)\,\cos(S-C)}}}\end{aligned}}} where a, b, c are the angular lengths (measure of central angle, arc lengths normalized to a sphere of unit radius) of the sides opposite angles A, B, C respectively, and S = 1 2 ( A + B + C ) {\displaystyle S={\tfrac {1}{2}}(A+B+C)} is half the sum of the angles.