Formeln Trigonometrie

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Schiefwinkliges Dreieck:

Formeln Trigonometrie.bmp

Sinussatz:

\({a \over \sin \alpha} = {b \over \sin \beta} = {c \over \sin \gamma} = 2r \)


Kosinussatz:


\(c^2 =a^2+b^2-2ab \cos \gamma\)


Höhe:


\(h = a \sin \beta = b \sin \alpha \)



Zusammenhang zwischen Sinus, Kosinus und Tangens:

\( \cos \alpha = \sin ( a + {\pi \over 2})\)

\(\sin^2 \alpha = \cos^2 \alpha = 1\)


\(\sin \alpha = + \sqrt{1-\cos^2 \alpha}\)

\(\cos \alpha = + \sqrt{1-\sin^2 \alpha}\)

\(\tan \alpha = {\sin \alpha \over cos\ \alpha} \)

\({1 \over \sqrt{ 1 + \tan^2 x}} = \pm \cos x \)




Additionstheoreme:

\(\sin (\alpha + \beta) = \sin \alpha *\cos \beta + \cos \alpha * \sin \beta\)

\(\cos (\alpha + \beta) = \cos \alpha *\cos \beta - \sin \alpha * \sin \beta\)

\(\sin (\alpha - \beta) = \sin \alpha *\cos \beta - \cos \alpha * \sin \beta\)

\(\cos (\alpha - \beta) = \cos \alpha *\cos \beta + \sin \alpha * \sin \beta\)



Produkte von Winkelfunktionen:

\(\sin \alpha * \sin \beta = {1 \over 2} \cos (\alpha - \beta)- {1 \over 2}\cos (\alpha + \beta) \)

\(\sin \alpha * \cos \beta = {1 \over 2} \sin (\alpha - \beta)- {1 \over 2}\sin (\alpha + \beta) \)