Mollweide's formula

Figure 1 – A triangle. The angles α, β, and γ are respectively opposite the sides a, b, and c.

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

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

and

{\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.

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

Mollweide's formula

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