What is the DeMoivre's theorem used for?

Question

7645 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-07-25T10:54:41

More of the cases, to find expresions for $sinnx$ or $cosnx$ as function of $sinx$ and $cosx$ and their powers. See below

Moivre's theorem says that $(cosx+isinx)^n=cosnx+isinnx$

An example ilustrates this. Imagine that we want to find an expresion for $cos^3x$ . Then

$(cosx+isinx)^3=cos3x+isin3x$ by De Moivre's theorem

By other hand applying binomial Newton's theorem, we have

$(cosx+isinx)^3=cos^3x+3icos^2xsinx+3i^2cosxsin^2x+i^3sin^3x=cos^3x-3cosxsin^2x+(3cos^2xsinx-sin^3x)i$

Then, equalizing both expresions as conclusion we have

$cos3x=cos^3x-3cosxsin^2x$
$sin3x=3cos^2xsinx-sin^3x$