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, named after Karl Mollweide, is a set of two relationships between sides and angles in a triangle.
. Substitute into the left denominator and using difference formula on the right denominator, 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 theorem .
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 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 .
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 .