Mollweide's formula

In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle.

A variant in more geometrical style was first published by Isaac Newton in 1707 and then by Friedrich Wilhelm von Oppel in 1746. Thomas Simpson published the now-standard expression in 1748. Karl Mollweide republished the same result in 1808 without citing those predecessors.

It can be used to check the consistency of solutions of triangles.

Let ${\displaystyle a,}$ ${\displaystyle b,}$ and ${\displaystyle c}$ be the lengths of the three sides of a triangle. Let ${\displaystyle \alpha ,}$ ${\displaystyle \beta ,}$ and ${\displaystyle \gamma }$ be the measures of the angles opposite those three sides respectively. Mollweide's formulas are

${\displaystyle {\begin{aligned}{\frac {a+b}{c}}={\frac {\cos {\tfrac {1}{2}}(\alpha -\beta )}{\sin {\tfrac {1}{2}}\gamma }},\\[10mu]{\frac {a-b}{c}}={\frac {\sin {\tfrac {1}{2}}(\alpha -\beta )}{\cos {\tfrac {1}{2}}\gamma }}.\end{aligned}}}$