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Derivatives, symmetries and inverses of trigonometric functions

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.

Image

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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