Mollweid formulas

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Original publication from 1808
Names of the sides and angles

The minor grazing rule formulas , named after the German mathematician and astronomer Karl Mollweide are trigonometric formulas for arbitrary triangles apply. Isaac Newton discovered these relationships a century earlier.

Derivation  

Sine law :

(1)
(2)

Sinus identities :

(3)
(4)

Sine addition theorem for double angles :

(5)

Angle sum in the triangle and transition to the complementary angle :

(6)
(7)

Addition of (1) and (2), application of (3) and (5), truncation using (6):

Subtract (1) - (2), apply (4) and (5), truncate using (7):

Multiplication with the common denominator gives the formulas given. The other two formulas, which contain a sum or a difference between two sides, are created through cyclic substitution of the side and angle names

source

  • CB Mollweide: Additions to plane and spherical trigonometry. In: Monthly correspondence for conveyance of the earth and sky customer, 1808, pages 394–400.