Mollweid formulas

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.

${\displaystyle (b+c)\sin {\frac {\alpha }{2}}=a\cos {\frac {\beta -\gamma }{2}}}$

${\displaystyle (b-c)\cos {\frac {\alpha }{2}}=a\sin {\frac {\beta -\gamma }{2}}}$

${\displaystyle (c+a)\sin {\frac {\beta }{2}}=b\cos {\frac {\gamma -\alpha }{2}}}$

${\displaystyle (c-a)\cos {\frac {\beta }{2}}=b\sin {\frac {\gamma -\alpha }{2}}}$

${\displaystyle (a+b)\sin {\frac {\gamma }{2}}=c\cos {\frac {\alpha -\beta }{2}}}$

${\displaystyle (a-b)\cos {\frac {\gamma }{2}}=c\sin {\frac {\alpha -\beta }{2}}}$

Derivation

Sine law : ${\displaystyle {b \over a}={\sin(\beta ) \over \sin(\alpha )};\quad }$(1) ${\displaystyle {c \over a}={\sin(\gamma ) \over \sin(\alpha )};\quad }$(2) Sinus identities : ${\displaystyle \sin(\beta )+\sin(\gamma )=2\cdot \sin({\beta +\gamma \over 2})\cdot \cos({\beta -\gamma \over 2});\quad }$(3) ${\displaystyle \sin(\beta )-\sin(\gamma )=2\cdot \cos({\beta +\gamma \over 2})\cdot \sin({\beta -\gamma \over 2});\quad }$(4) Sine addition theorem for double angles : ${\displaystyle \sin(\alpha )=\sin(2\cdot {\alpha \over 2})=2\cdot \sin({\alpha \over 2})\cdot \cos({\alpha \over 2});\quad }$(5) Angle sum in the triangle and transition to the complementary angle : ${\displaystyle \sin({\beta +\gamma \over 2})=\sin({180^{\circ }-\alpha \over 2})=\sin(90^{\circ }-{\alpha \over 2})=\cos({\alpha \over 2});\quad }$(6) ${\displaystyle \cos({\beta +\gamma \over 2})=\cos({180^{\circ }-\alpha \over 2})=\cos(90^{\circ }-{\alpha \over 2})=\sin({\alpha \over 2});\quad }$(7) Addition of (1) and (2), application of (3) and (5), truncation using (6): ${\displaystyle {b+c \over a}={\sin(\beta )+\sin(\gamma ) \over \sin(\alpha )}=}$ ${\displaystyle {\frac {2\cdot \sin({\beta +\gamma \over 2})\cdot \cos({\beta -\gamma \over 2})}{2\cdot \sin({\alpha \over 2})\cdot \cos({\alpha \over 2})}}=}$ ${\displaystyle {\frac {\cos({\beta -\gamma \over 2})}{\sin({\alpha \over 2})}};}$ Subtract (1) - (2), apply (4) and (5), truncate using (7): ${\displaystyle {b-c \over a}={\sin(\beta )-\sin(\gamma ) \over \sin(\alpha )}={\frac {2\cdot \cos({\beta +\gamma \over 2})\cdot \sin({\beta -\gamma \over 2})}{2\cdot \sin({\alpha \over 2})\cdot \cos({\alpha \over 2})}}={\frac {\sin({\beta -\gamma \over 2})}{\cos({\alpha \over 2})}};}$ 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.