Trigonometrie
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(→trigonometrische Äquivalenzen) |
Current revision (21:47, 16 September 2008) (view source) (→Produkt -> Summen oder Differenz) |
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| - | == | + | == Definitionen == |
| - | {| | + | {|width=400 |
| + | |<math>tan(\varphi) = \frac {\sin(\varphi)} {\cos(\varphi)}</math> | ||
| + | |<math>cot(\varphi) = \frac {\cos(\varphi)} {\sin(\varphi)}</math> | ||
| + | |} | ||
| + | |||
| + | == Symmetrieen == | ||
| + | {|width=400 | ||
| + | |<math>\sin(\varphi+k\cdot 360^\circ) = \sin(\varphi)</math> | ||
| + | |<math>\cos(\varphi+k\cdot 360^\circ) = \cos(\varphi)</math> | ||
| + | |- | ||
| + | |<math>\tan(\varphi+k\cdot 180^\circ) = \tan(\varphi)</math> | ||
| + | |<math>\cot(\varphi+k\cdot 180^\circ) = \cot(\varphi)</math> | ||
| + | |- | ||
|<math>\sin(-\varphi) = -\sin(\varphi)</math> | |<math>\sin(-\varphi) = -\sin(\varphi)</math> | ||
|<math>\cos(-\varphi) = \quad\cos(\varphi)</math> | |<math>\cos(-\varphi) = \quad\cos(\varphi)</math> | ||
| Line 7: | Line 19: | ||
|<math>\tan(-\varphi) = -\tan(\varphi)</math> | |<math>\tan(-\varphi) = -\tan(\varphi)</math> | ||
|<math>\cot(-\varphi) = -\cot(\varphi)</math> | |<math>\cot(-\varphi) = -\cot(\varphi)</math> | ||
| + | |} | ||
| + | |||
| + | == trigonometrische Äquivalenzen == | ||
| + | |||
| + | {|width=400 | ||
| + | |<math>\sin(90^\circ-\varphi) = \cos(\varphi)</math> | ||
| + | |<math>\cos(90^\circ-\varphi) = \sin(\varphi)</math> | ||
|- | |- | ||
| - | |<math></math> | + | |<math>\tan(90^\circ-\varphi) = \cot(\varphi)</math> |
| + | |<math>\cot(90^\circ-\varphi) = \tan(\varphi)</math> | ||
|} | |} | ||
<math>\sin^2(\varphi) + \cos^2(\varphi) = 1</math> | <math>\sin^2(\varphi) + \cos^2(\varphi) = 1</math> | ||
| - | + | ||
<math>\tan(\varphi) \cdot \cot(\varphi) = 1, \quad \varphi \ne k\cdot 90 ^\circ</math> | <math>\tan(\varphi) \cdot \cot(\varphi) = 1, \quad \varphi \ne k\cdot 90 ^\circ</math> | ||
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<math>1 + \cot^2(\varphi) = \frac 1 \sin^2(\varphi), \quad \varphi \ne k\cdot 180^\circ</math> | <math>1 + \cot^2(\varphi) = \frac 1 \sin^2(\varphi), \quad \varphi \ne k\cdot 180^\circ</math> | ||
| + | |||
| + | == Additionstheoreme == | ||
| + | |||
| + | <math>\sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta</math> | ||
| + | |||
| + | <math>\cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta</math> | ||
| + | |||
| + | <math>\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta} {1 - \tan\alpha\, \tan\beta}</math> | ||
| + | |||
| + | <math>\sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta</math> | ||
| + | |||
| + | <math>\cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta</math> | ||
| + | |||
| + | <math>\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta} {1 + \tan\alpha\, \tan\beta}</math> | ||
| + | |||
| + | == doppelte Winkel == | ||
| + | |||
| + | <math>\sin 2 \alpha = 2 \sin\alpha\, \cos\alpha</math> | ||
| + | |||
| + | <math>\cos 2 \alpha = \cos^2\alpha 2 - \sin^2\alpha = 2 cos^2\alpha - 1 = 1 - 2 \sin^2\alpha</math> | ||
| + | |||
| + | <math>\tan 2 \alpha = \frac {2 \tan\alpha} {1 - \tan^2\alpha}</math> | ||
| + | |||
| + | == halbe Winkel == | ||
| + | |||
| + | {|width=400 | ||
| + | |<math>\sin^2\frac\alpha 2 = \frac 1 2 (1 - \cos\alpha)</math> | ||
| + | |<math>1 - \cos 2\alpha = 2 \sin^2\alpha</math> | ||
| + | |- | ||
| + | |<math>\cos^2\frac\alpha 2 = \frac 1 2 (1 + \cos\alpha)</math> | ||
| + | |<math>1 + \cos 2\alpha = 2 \cos^2\alpha</math> | ||
| + | |- | ||
| + | |<math>\tan^2\frac\alpha 2 = \frac {1 - \cos\alpha}{1 + \cos\alpha}</math> | ||
| + | |} | ||
| + | |||
| + | == Summen und Differenzen -> Produkt == | ||
| + | |||
| + | <math>\sin\alpha + \sin\beta = 2 \sin\frac{\alpha+\beta} 2 \cos\frac{\alpha-\beta} 2</math> | ||
| + | |||
| + | <math>\sin\alpha - \sin\beta = 2 \cos\frac{\alpha+\beta} 2 \sin\frac{\alpha-\beta} 2</math> | ||
| + | |||
| + | <math>\cos\alpha + \cos\beta = 2 \cos\frac{\alpha+\beta} 2 \cos\frac{\alpha-\beta} 2</math> | ||
| + | |||
| + | <math>\cos\alpha - \cos\beta = -2 \sin\frac{\alpha+\beta} 2 \sin\frac{\alpha-\beta} 2</math> | ||
| + | |||
| + | == Produkt -> Summen oder Differenz == | ||
| + | |||
| + | <math>2\,\sin\alpha \sin\beta = \cos(\alpha-\beta} - \cos(\alpha+\beta)</math> | ||
| + | |||
| + | <math>2\,\cos\alpha \cos\beta = \cos(\alpha-\beta} + \cos(\alpha+\beta)</math> | ||
| + | |||
| + | <math>2\,\sin\alpha \cos\beta = \sin(\alpha-\beta} + \sin(\alpha+\beta)</math> | ||
[[category:Mathematik]] | [[category:Mathematik]] | ||
Current revision
Contents |
Definitionen
Symmetrieen
trigonometrische Äquivalenzen
Additionstheoreme
doppelte Winkel
halbe Winkel
Summen und Differenzen -> Produkt
Produkt -> Summen oder Differenz
