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Derivatives, symmetries and inverses of trigonometric functions

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There are certain relationships between the trigonometric functions with regard to the derivative, symmetry and inverse function, which you can find in the table below.

Sinus

(sin⁡(x))′=cos⁡(x)(\sin(x))'=\cos(x)

Punktsymmetrisch zum Ursprung

Arkussinus:

Kosinus

(cos⁡(x))′=−sin⁡(x)(\cos(x))' = -\sin(x)

Achsensymmetrisch zur yy-Achse

Arkuskosinus:

Tangens

(tan⁡(x))′=1+tan⁡2(x)=1cos⁡2(x)(\tan(x))' = 1 + \tan^2(x) = \dfrac{1}{\cos^2(x)}

Punktsymmetrisch zum Ursprung:

Arkustangens:

Beispiel

Leite die Funktion  f(x)=cos⁡(x)−2sin⁡(x) ~f(x)=\cos(x)-2\sin(x)~ ab.

f′(x)=(cos⁡(x))′−2(sin⁡(x))′f'(x)=\left( \cos(x) \right)' -2 \left(\sin(x) \right)'

Schaue in der obigen Abbildung nach, was die Ableitung der Sinus- beziehungsweise Kosinusfunktion ist.

f′(x)=−sin⁡(x)−2cos⁡(x)f'(x)=-\sin(x)-2\cos(x)

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