de Moivre's formula

Builds on Euler's formula Product of exponentials

Statement

For every real number $x$ and every integer $n$ , $(\cos x + i \sin x)^n = \cos(n x) + i \sin(n x).$

Remarks

Proof: by Euler's formula, the left side is $(e^{i x})^n = e^{i n x}$ , which is the right side. For $n = 2$ , comparing real and imaginary parts gives $\cos 2x = \cos^2 x - \sin^2 x$ and $\sin 2x = 2 \sin x \cos x$ . For non-integer exponents $n$ , the formula fails in general.

Reading path

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Where this leads

5 pages build on this one.

Tag	`03MTBVG3VE`
URI	`m2://03MTBVG3VE`
Kind	statement, theorem
Writing	W1 stated
MSC	33B10
Status	This copy of the site has no build data. Run `m2 status 03MTBVG3VE`.
Uses	Euler's formula, Product of exponentials
Same as	Wikidata Q190556 Wikipedia De Moivre's formula

de Moivre's formula

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

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