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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 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

and

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

Contents

DerivationEdit

First Mollweide's formulaEdit

Method 1: Law of sines and angle bisector theoremEdit

 
Figure 2 - Derivation of the first Mollweide's formula

The first Mollweide's formula can be derived from the law of sines and the angle bisector theorem.

In Figure 2, CD is the angle bisector of  . From the angle bisector theorem,   but  . Rearranging,   .   and  . Applying the law of sines on ΔACD,

 . Substitute   into the left denominator and using difference formula on the right numerator, after simplifying, we get  . Substitute   into the right numerator and upon simplifying, we get  . Applying sum formula on the right numerator, we get the first Mollweide's formula  .

Method 2: Law of cotangentsEdit

Method 3: Law of sines and cosinesEdit

Applying the law of sines on Figure 1,   and  . Applying the law of cosines,  . Rearranging,  . This is the beginning to get both the formulas. To get the first Mollweide's formula, rearranging further,  . Substitute   and   in the right hand side of the equation and simplifying,  . Substitute   and use the sum and difference formulas and simplifying the right numerator,  . Using difference formula for sine on the right numerator, the equation becomes  . Using double angle formula for the right numerator and sum-to-product formula for the right denominator, the equation becomes   . Simplifying,  . Substitute  , it becomes  . Using the complementary angle theorem on the right denominator, we arrive at the first Mollweide's formula  .

Second Mollweide's formulaEdit

Method 1: Law of cotangentsEdit

Method 2: Law of sines and cosinesEdit

The method is similar to the one for the first Mollweide's formula. We rearrange   to get  . The right numerator is the same as for the first formula, that is  . However, the right denominator is an addition instead of a subtraction, that is  . Using the sum-to-product formula, it changes to  . Thus,  . Simplifying and substitute   in the right denominator, we get  . Using the complementary angle theorem on the right denominator, we arrive at the second Mollweide's formula  .

See alsoEdit

ReferencesEdit

  1. ^ Ernest Julius Wilczynski, Plane Trigonometry and Applications, Allyn and Bacon, 1914, page 102
  2. ^ Michael Sullivan, Trigonometry, Dellen Publishing Company, 1988, page 243.
  3. ^ Ernest Julius Wilczynski, Plane Trigonometry and Applications, Allyn and Bacon, 1914, page 105

Additional readingEdit

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