Zurück zu Identitäten
1. sin ( α + β ) + sin ( α − β ) = 2 ⋅ sin α ⋅ cos β {\displaystyle 1.\quad \sin(\alpha +\beta )+\sin(\alpha -\beta )=2\cdot \sin \alpha \cdot \cos \beta } 2. sin ( α + β ) − sin ( α − β ) = 2 ⋅ cos α ⋅ sin β {\displaystyle 2.\quad \sin(\alpha +\beta )-\sin(\alpha -\beta )=2\cdot \cos \alpha \cdot \sin \beta } 3. cos ( α + β ) + cos ( α − β ) = 2 ⋅ cos α ⋅ cos β {\displaystyle 3.\quad \cos(\alpha +\beta )+\cos(\alpha -\beta )=2\cdot \cos \alpha \cdot \cos \beta } 4. cos ( α + β ) − cos ( α − β ) = − 2 ⋅ sin α ⋅ sin β {\displaystyle 4.\quad \cos(\alpha +\beta )-\cos(\alpha -\beta )=-2\cdot \sin \alpha \cdot \sin \beta }
Die Formeln folgen aus den Additionstheoremen sin ( α + β ) = sin α ⋅ cos β + cos α ⋅ sin β {\displaystyle \sin(\alpha +\beta )=\sin \alpha \cdot \cos \beta +\cos \alpha \cdot \sin \beta } sin ( α − β ) = sin α ⋅ cos β − cos α ⋅ sin β {\displaystyle \sin(\alpha -\beta )=\sin \alpha \cdot \cos \beta -\cos \alpha \cdot \sin \beta } cos ( α + β ) = cos α ⋅ cos β − sin α ⋅ sin β {\displaystyle \cos(\alpha +\beta )=\cos \alpha \cdot \cos \beta -\sin \alpha \cdot \sin \beta } cos ( α − β ) = cos α ⋅ cos β + sin α ⋅ sin β {\displaystyle \cos(\alpha -\beta )=\cos \alpha \cdot \cos \beta +\sin \alpha \cdot \sin \beta }
1. sin α + sin β = 2 ⋅ sin α + β 2 ⋅ cos α − β 2 {\displaystyle 1.\quad \sin \alpha +\sin \beta =2\cdot \sin {\frac {\alpha +\beta }{2}}\cdot \cos {\frac {\alpha -\beta }{2}}} 2. sin α − sin β = 2 ⋅ cos α + β 2 ⋅ sin α − β 2 {\displaystyle 2.\quad \sin \alpha -\sin \beta =2\cdot \cos {\frac {\alpha +\beta }{2}}\cdot \sin {\frac {\alpha -\beta }{2}}} 3. cos α + cos β = 2 ⋅ cos α + β 2 ⋅ cos α − β 2 {\displaystyle 3.\quad \cos \alpha +\cos \beta =2\cdot \cos {\frac {\alpha +\beta }{2}}\cdot \cos {\frac {\alpha -\beta }{2}}} 4. cos α − cos β = − 2 ⋅ sin α + β 2 ⋅ sin α − β 2 {\displaystyle 4.\quad \cos \alpha -\cos \beta =-2\cdot \sin {\frac {\alpha +\beta }{2}}\cdot \sin {\frac {\alpha -\beta }{2}}}
Die Formeln nach Simpson folgen aus den Formeln nach Werner durch die Substitution [ x = α + β y = α − β ] {\displaystyle {\begin{bmatrix}x=\alpha +\beta \\y=\alpha -\beta \end{bmatrix}}} bzw. [ α = ( x + y ) / 2 β = ( x − y ) / 2 ] {\displaystyle {\begin{bmatrix}\alpha =(x+y)/2\\\beta =(x-y)/2\end{bmatrix}}} .
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