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Question

How do you use de Moivre's theorem to find roots?


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Solution

Find the root by De Moivre’s theorem:

De Moivre’s theorem states that:

Suppose z=rcosθ+i(rsinθ) is a complex number and n be a positive integer such that

zn=[r(cosθ+isinθ)]n

zn=rn(cosnθ+isinnθ)

For example;

Let a complex number be cosx+isinx3

Here, n=3 and r=1

Now, Apply De Moivre’s theorem to find the roots

rcosx+irsinxn=rncosnx+isinnx

cosx+isinx3=1cos3x+isin3x=cos3x+isin3x

Hence, De Moivre’s theorem is used to find the roots of complex numbers.


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