Half-side formula

On a unit sphere, the half-side formulas are[2]

${\displaystyle {\begin{aligned}\tan \left({\frac {a}{2}}\right)&=R\cos(S-A)\\[8pt]\tan \left({\frac {b}{2}}\right)&=R\cos(S-B)\\[8pt]\tan \left({\frac {c}{2}}\right)&=R\cos(S-C)\end{aligned}}}$

where

• a, b, c are the lengths of the sides respectively opposite angles A, B, C,
• ${\displaystyle S={\frac {1}{2}}(A+B+C)}$ is half the sum of the angles, and
• ${\displaystyle R={\sqrt {\frac {-\cos S}{\cos(S-A)\cos(S-B)\cos(S-C)}}}.}$

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

To generalize to a sphere of arbitrary radius r, the lengths a,b,c must be replaced with

• ${\displaystyle a\rightarrow a/r}$
• ${\displaystyle b\rightarrow b/r}$
• ${\displaystyle c\rightarrow c/r}$

so that a,b,c all have length scales, instead of angular scales.

• Spherical law of cosines
• Law of haversines