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

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 formulas are

: \begin
\frac c
= \frac

, \\ 0mu
\frac c
= \frac

.
\end

Relation to other trigonometric identities

Because in a planar triangle $\tfrac12\gamma = \tfrac12\pi - \tfrac12(\alpha + \beta),$ these identities can alternately be written in a form in which they are more clearly a limiting case of Napier's analogies for spherical triangles,

: \begin
\frac c
&= \frac

, \\ 0mu
\frac c
&= \frac

.
\end

Dividing one by the other to eliminate $c$ results in the law of tangents,

: $\begin \frac = \frac . \end$

In terms of half-angle tangents alone, Mollweide's formula can be written as

: \begin
\frac c
&= \frac

, \\ 0mu
\frac c
&= \frac

,
\end

or equivalently

: \begin
\tan\tfrac12\alpha\,\tan\tfrac12\beta
&= \frac

, \\ 0mu
\frac

&= \frac

.
\end

Multiplying the respective sides of these identities gives one half-angle tangent in terms of the three sides,

: $\bigl(\tan\tfrac12\alpha\bigr)^2 = \frac .$

which becomes the law of cotangents after taking the square root,

: $\frac = \frac = \frac = \sqrt,$

where $s = \tfrac12(a + b + c)$ is the semiperimeter.

The identities can also be proven equivalent to the law of sines and law of cosines.

References

Further reading

* H. Arthur De Kleine, "Proof Without Words: Mollweide's Equation", '' Mathematics Magazine'', volume 61, number 5, page 281, December, 1988.

{{DEFAULTSORT:Mollweide's Formula
Trigonometry
Theorems about triangles