# 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

$$\frac{{d}^{n}}{d{x}^{n}}\mathrm{sin}x=\mathrm{sin}(x+n\cdot \frac{\pi}{2})$$ |

and

$$\frac{{d}^{n}}{d{x}^{n}}\mathrm{cos}x=\mathrm{cos}(x+n\cdot \frac{\pi}{2}),$$ |

where $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots}$ (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

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

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 |