de Moivre identity
From the Euler relation
eiθ=cosθ+isinθ |
it follows that
eiθ⋅n | =(eiθ)n | ||
cosnθ+isinnθ | =(cosθ+isinθ)n |
where n∈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,
cos2θ+isin2θ=(cosθ+isinθ)2=cos2θ+2isinθcosθ-sin2θ. |
Since the imaginary parts and real parts on each side must be equal, we must have
cos2θ=cos2θ-sin2θ |
and
sin2θ=2sinθcosθ. |
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 |