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.^[1]

For a triangle ${\displaystyle \triangle ABC}$ on a sphere, the half-side formula is^[2] $${\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}a&={\sqrt {\frac {-\cos(S)\,\cos(S-A)}{\cos(S-B)\,\cos(S-C)}}}\end{aligned}}}$$

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 ${\displaystyle S={\tfrac {1}{2}}(A+B+C)}$ is half the sum of the angles. Two more formulas can be obtained for ${\displaystyle b}$ and ${\displaystyle c}$ by permuting the labels ${\displaystyle A,B,C.}$

The polar dual relationship for a spherical triangle is the half-angle formula,

$${\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}A&={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin(s)\,\sin(s-a)}}}\end{aligned}}}$$

where semiperimeter ${\displaystyle s={\tfrac {1}{2}}(a+b+c)}$ is half the sum of the sides. Again, two more formulas can be obtained by permuting the labels ${\displaystyle A,B,C.}$