# higher order derivatives of sine and cosine

One may consider the sine and cosine either as real (http://planetmath.org/RealFunction) or complex functions.  In both cases they are everywhere smooth, having the derivatives of all orders (http://planetmath.org/OrderOfDerivative) in every point.  The formulae

 $\frac{d^{n}}{dx^{n}}\sin{x}\;=\;\sin{(x+n\!\cdot\!\frac{\pi}{2})}$

and

 $\frac{d^{n}}{dx^{n}}\cos{x}\;=\;\cos{(x+n\!\cdot\!\frac{\pi}{2})},$

where  $n=0,\,1,\,2,\,\ldots$ (the derivative of the $0^{\mathrm{th}}$ order means the function itself), can be proven by induction on $n$.  Another possibility is to utilize Euler’s formula, obtaining

 $\frac{d^{n}}{dx^{n}}\cos{x}+i\frac{d^{n}}{dx^{n}}\sin{x}\;=\;\frac{d^{n}}{dx^{% n}}e^{ix}\;=\;e^{ix}i^{n}\;=\;e^{ix+in\frac{\pi}{2}}\;=\;\cos{(x+n\!\cdot\!% \frac{\pi}{2})}+i\sin{(x+n\!\cdot\!\frac{\pi}{2})};$

here one has to compare the real (http://planetmath.org/ComplexFunction) and imaginary parts – supposing that $x$ is real.

 Title higher order derivatives of sine and cosine Canonical name HigherOrderDerivativesOfSineAndCosine Date of creation 2013-03-22 14:45:16 Last modified on 2013-03-22 14:45:16 Owner pahio (2872) Last modified by pahio (2872) Numerical id 13 Author pahio (2872) Entry type Derivation Classification msc 26B05 Classification msc 46G05 Classification msc 26A24 Related topic FractionalDifferentiation Related topic HigherOrderDerivatives Related topic ExampleOfTaylorPolynomialsForSinX Related topic CosineAtMultiplesOfStraightAngle