trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies ...

, 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 Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...

in 1707 and then by in 1746.

Thomas Simpson Thomas Simpson FRS (20 August 1710 – 14 May 1761) was a British mathematician and inventor known for the eponymous Simpson's rule to approximate definite integrals. The attribution, as often in mathematics, can be debated: this rule had been ...

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 :

\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, :

\frac c &= \frac . \end

Dividing one by the other to eliminate

c

results in the

law of tangents In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, , , and are the lengths of the three sides of the triangle, and , , ...

, :

\begin
\frac
= \frac
  
  .
\end

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

\frac c &= \frac , \end

or equivalently :

\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 In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate na ...

. The identities can also be proven equivalent to the

law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and ar ...

and

law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...

Relation to other trigonometric identities

References

Further reading