Mollweide's formula

In trigonometry, Mollweide's formula, sometimes referred to in older texts as Mollweide's equations,[1] named after Karl Mollweide, is a set of two relationships between sides and angles in a triangle.[2]

It can be used to check the consistency of solutions of triangles.[3]

Let a, b, and c be the lengths of the three sides of a triangle. Let α, β, and γ be the measures of the angles opposite those three sides respectively. Mollweide's formula states that

\( {\displaystyle {\frac {a+b}{c}}={\frac {\cos \left({\frac {\alpha -\beta }{2}}\right)}{\sin \left({\frac {\gamma }{2}}\right)}}} \)

and

\( {\displaystyle {\frac {a-b}{c}}={\frac {\sin \left({\frac {\alpha -\beta }{2}}\right)}{\cos \left({\frac {\gamma }{2}}\right)}}.} \)

Each of these identities uses all six parts of the triangle—the three angles and the lengths of the three sides.
See also

Law of sines
Law of cosines
Law of tangents
Law of cotangents

References

Ernest Julius Wilczynski, Plane Trigonometry and Applications, Allyn and Bacon, 1914, page 102
Michael Sullivan, Trigonometry, Dellen Publishing Company, 1988, page 243.

Ernest Julius Wilczynski, Plane Trigonometry and Applications, Allyn and Bacon, 1914, page 105