Trigonometrie

From SkyWiki

Revision as of 20:57, 16 September 2008 (view source) Martin (Talk | contribs) (New page: == trigonometrische Äquivalenzen == <math>\sin(-\phi) = -\sin(phi)</math> <math>\cos(-\phi) = \quad\cos(phi)</math> category:Mathematik) ← Previous diff		Current revision (21:47, 16 September 2008) (view source) Martin (Talk | contribs) (→Produkt -> Summen oder Differenz)
(21 intermediate revisions not shown.)
Line 1:		Line 1:
		+	== 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>\cos(-\varphi) = \quad\cos(\varphi)</math>
		+	|-
		+	|<math>\tan(-\varphi) = -\tan(\varphi)</math>
		+	|<math>\cot(-\varphi) = -\cot(\varphi)</math>
		+	|}
		+
	== trigonometrische Äquivalenzen ==		== trigonometrische Äquivalenzen ==
-	<math>\sin(-\phi) = -\sin(phi)</math>	+	{|width=400
-	<math>\cos(-\phi) = \quad\cos(phi)</math>	+	|<math>\sin(90^\circ-\varphi) = \cos(\varphi)</math>
		+	|<math>\cos(90^\circ-\varphi) = \sin(\varphi)</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>\tan(\varphi) \cdot \cot(\varphi) = 1, \quad \varphi \ne k\cdot 90 ^\circ</math>
		+
		+	<math>1 + \tan^2(\varphi) = \frac 1 \cos^2(\varphi), \quad \varphi \ne (2k+1)\cdot 90^\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 1 Definitionen 2 Symmetrieen 3 trigonometrische Äquivalenzen 4 Additionstheoreme 5 doppelte Winkel 6 halbe Winkel 7 Summen und Differenzen -> Produkt 8 Produkt -> Summen oder Differenz

Definitionen

Symmetrieen

trigonometrische Äquivalenzen

Additionstheoreme

doppelte Winkel

halbe Winkel

Summen und Differenzen -> Produkt

Produkt -> Summen oder Differenz