# de Moivre identity

From the Euler relation

 $e^{i\theta}=\cos\theta+i\sin\theta$

it follows that

 $\displaystyle e^{i\theta\cdot n}$ $\displaystyle=(e^{i\theta})^{n}$ $\displaystyle\cos n\theta+i\sin n\theta$ $\displaystyle=(\cos\theta+i\sin\theta)^{n}$

where $n\in\mathbb{Z}$. This is called de Moivre’s formula, and besides being generally useful, it’s a convenient way to remember double- (and higher-multiple-) angle formulas. For example,

 $\cos 2\theta+i\sin 2\theta=(\cos\theta+i\sin\theta)^{2}=\cos^{2}\theta+2i\sin% \theta\cos\theta-\sin^{2}\theta.$

Since the imaginary parts and real parts on each side must be equal, we must have

 $\cos 2\theta=\cos^{2}\theta-\sin^{2}\theta$

and

 $\sin 2\theta=2\sin\theta\cos\theta.$
 Title de Moivre identity Canonical name DeMoivreIdentity Date of creation 2013-03-22 12:20:45 Last modified on 2013-03-22 12:20:45 Owner Daume (40) Last modified by Daume (40) Numerical id 11 Author Daume (40) Entry type Theorem Classification msc 12E10 Synonym de Moivre’s theorem Synonym de Moivre’s formula Related topic EulerRelation Related topic DoubleAngleIdentity Related topic ArgumentOfProductAndSum Related topic ArgumentOfProductAndQuotient