Mollweide's formula

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

A variant of one of the expressions in more geometrical style was first published by Isaac Newton in 1707. Both relations were published by Friedrich Wilhelm von Oppel in 1746, and then by Thomas Simpson 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

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