de Moivre identity

From the Euler relation

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

it follows that

$e^{i\theta \cdot n}$ $={(e^{i\theta })}^n$

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

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