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 ${\displaystyle A,B,C.}$

The same relationships can be written as rational equations of half-tangents (tangents of half-angles). If ${\displaystyle t_{a}=\tan {\tfrac {1}{2}}a,}$ ${\displaystyle t_{b}=\tan {\tfrac {1}{2}}b,}$ ${\displaystyle t_{c}=\tan {\tfrac {1}{2}}c,}$${\displaystyle t_{A}=\tan {\tfrac {1}{2}}A,}$ ${\displaystyle t_{B}=\tan {\tfrac {1}{2}}B,}$ and ${\displaystyle t_{C}=\tan {\tfrac {1}{2}}C,}$ then the half-side formula is equivalent to:

$${\displaystyle {\begin{aligned}t_{a}^{2}&={\frac {{\bigl (}t_{B}t_{C}+t_{C}t_{A}+t_{A}t_{B}-1{\bigr )}{\bigl (}{-t_{B}t_{C}+t_{C}t_{A}+t_{A}t_{B}+1}{\bigr )}}{{\bigl (}t_{B}t_{C}-t_{C}t_{A}+t_{A}t_{B}+1{\bigr )}{\bigl (}t_{B}t_{C}+t_{C}t_{A}-t_{A}t_{B}+1{\bigr )}}}.\end{aligned}}}$$

and the half-angle formula is equivalent to:

$${\displaystyle {\begin{aligned}t_{A}^{2}&={\frac {{\bigl (}t_{a}-t_{b}+t_{c}+t_{a}t_{b}t_{c}{\bigr )}{\bigl (}t_{a}+t_{b}-t_{c}+t_{a}t_{b}t_{c}{\bigr )}}{{\bigl (}t_{a}+t_{b}+t_{c}-t_{a}t_{b}t_{c}{\bigr )}{\bigl (}{-t_{a}+t_{b}+t_{c}+t_{a}t_{b}t_{c}}{\bigr )}}}.\end{aligned}}}$$

• Spherical law of cosines
• Law of haversines