Formelsammlung Mathematik: Identitäten: Prosthaphäretische Formeln

Formeln nach Werner

$1.\quad \sin(\alpha +\beta )+\sin(\alpha -\beta )=2\cdot \sin \alpha \cdot \cos \beta$

$2.\quad \sin(\alpha +\beta )-\sin(\alpha -\beta )=2\cdot \cos \alpha \cdot \sin \beta$

$3.\quad \cos(\alpha +\beta )+\cos(\alpha -\beta )=2\cdot \cos \alpha \cdot \cos \beta$

$4.\quad \cos(\alpha +\beta )-\cos(\alpha -\beta )=-2\cdot \sin \alpha \cdot \sin \beta$

Beweis

Die Formeln folgen aus den Additionstheoremen

$\sin(\alpha +\beta )=\sin \alpha \cdot \cos \beta +\cos \alpha \cdot \sin \beta$

$\sin(\alpha -\beta )=\sin \alpha \cdot \cos \beta -\cos \alpha \cdot \sin \beta$

$\cos(\alpha +\beta )=\cos \alpha \cdot \cos \beta -\sin \alpha \cdot \sin \beta$

$\cos(\alpha -\beta )=\cos \alpha \cdot \cos \beta +\sin \alpha \cdot \sin \beta$

Formeln nach Simpson

$1.\quad \sin \alpha +\sin \beta =2\cdot \sin {\frac {\alpha +\beta }{2}}\cdot \cos {\frac {\alpha -\beta }{2}}$

$2.\quad \sin \alpha -\sin \beta =2\cdot \cos {\frac {\alpha +\beta }{2}}\cdot \sin {\frac {\alpha -\beta }{2}}$

$3.\quad \cos \alpha +\cos \beta =2\cdot \cos {\frac {\alpha +\beta }{2}}\cdot \cos {\frac {\alpha -\beta }{2}}$

$4.\quad \cos \alpha -\cos \beta =-2\cdot \sin {\frac {\alpha +\beta }{2}}\cdot \sin {\frac {\alpha -\beta }{2}}$

Beweis

Die Formeln nach Simpson folgen aus den Formeln nach Werner durch die Substitution

${\begin{bmatrix}x=\alpha +\beta \\y=\alpha -\beta \end{bmatrix}}$ bzw. ${\begin{bmatrix}\alpha =(x+y)/2\\\beta =(x-y)/2\end{bmatrix}}$ .

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