Question

What is the DeMoivre's theorem used for?

Answer 1

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

Explanation:

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`