What is the DeMoivre's theorem used for?

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What is the DeMoivre's theorem used for?

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Answer 1 · 2018-07-25T10:54:41

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

Explanation:

Moivre's theorem says that ${(cos x + i sin x)}^{n} = cos n x + i sin n x$ $(cosx+isinx)^n=cosnx+isinnx$

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

${(cos x + i sin x)}^{3} = cos 3 x + i sin 3 x$ $(cosx+isinx)^3=cos3x+isin3x$ by De Moivre's theorem

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

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

Then, equalizing both expresions as conclusion we have

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

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What is the DeMoivre's theorem used for?

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