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 $$\triangle ABC$$ on a sphere, the half-side formula is $$\begin{align} \tan \tfrac12 a &= \sqrt{\frac{-\cos(S)\, \cos(S - A)} {\cos(S - B)\, \cos(S - C)} } \end{align}$$

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 $$S = \tfrac12 (A+B+ C)$$ is half the sum of the angles. Two more formulas can be obtained for $$b$$ and $$c$$ by permuting the labels $$A, B, C.$$

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

$$\begin{align} \tan \tfrac12 A &= \sqrt{\frac{\sin(s - b)\, \sin(s - c)} {\sin(s)\, \sin(s - a)} } \end{align}$$

where semiperimeter $$s = \tfrac12 (a + b + c)$$ is half the sum of the sides. Again, two more formulas can be obtained by permuting the labels $$A, B, C.$$

Half-tangent variant
The same relationships can be written as rational equations of half-tangents (tangents of half-angles). If $$t_a = \tan \tfrac12 a,$$ $$t_b = \tan \tfrac12 b,$$ $$t_c = \tan \tfrac12 c,$$$$t_A = \tan \tfrac12 A,$$ $$t_B = \tan \tfrac12 B,$$ and $$t_C = \tan \tfrac12 C,$$ then the half-side formula is equivalent to:

$$\begin{align} t_a^2 &= \frac{\bigl(t_Bt_C + t_Ct_A + t_At_B - 1\bigr)\bigl({-t_Bt_C + t_Ct_A + t_At_B + 1}\bigr)} {\bigl(t_Bt_C - t_Ct_A + t_At_B + 1\bigr)\bigl(t_Bt_C + t_Ct_A - t_At_B + 1\bigr)}. \end{align}$$

and the half-angle formula is equivalent to:

$$\begin{align} t_A^2 &= \frac{\bigl(t_a - t_b + t_c + t_at_bt_c\bigr)\bigl(t_a + t_b - t_c + t_at_bt_c\bigr)} {\bigl(t_a + t_b + t_c - t_at_bt_c\bigr)\bigl({-t_a + t_b + t_c + t_at_bt_c}\bigr)}. \end{align}$$