Mollweide's formula

In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle.^[1]^[2]

A variant in more geometrical style was first published by Isaac Newton in 1707 and then by Friedrich Wilhelm von Oppel [de] in 1746. Thomas Simpson published the now-standard expression in 1748. Karl Mollweide republished the same result in 1808 without citing those predecessors.^[3]

It can be used to check the consistency of solutions of triangles.^[4]

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

{\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}}

^ Wilczynski, Ernest Julius (1914), Plane Trigonometry and Applications, Allyn and Bacon, p. 102
^ Sullivan, Michael (1988), Trigonometry, Dellen, p. 243
^ Bradley, H. C.; Yamanouti, T.; Lovitt, W. V.; Archibald, R. C. (1921), "Discussions: Geometric Proofs of the Law of Tangents", American Mathematical Monthly, 28 (11–12): 440–443
^ Ernest Julius Wilczynski, Plane Trigonometry and Applications, Allyn and Bacon, 1914, page 105

[1] Wilczynski, Ernest Julius (1914), Plane Trigonometry and Applications, Allyn and Bacon, p. 102

[2] Sullivan, Michael (1988), Trigonometry, Dellen, p. 243

[3] Bradley, H. C.; Yamanouti, T.; Lovitt, W. V.; Archibald, R. C. (1921), "Discussions: Geometric Proofs of the Law of Tangents", American Mathematical Monthly, 28 (11–12): 440–443

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

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