Overview

De Moivre's Theorem


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From the definition of multiplication (of a complex number), it follows that (cos θ1+i sin θ1)(cos θ2+i sin θ2)=cos (θ1+θ2)+i sin (θ1+θ2). This leads to the following result known as De Moivre's Theorem, which is crucial to any consideration of the powers zn of a complex number z:

Theorem

For all positive integers n, (cos θ+i sin θ)n=cos +i sin .

The result is also true for negative (and zero) integer values of n, and this may be considered as either included in or forming an extension of De Moivre's Theorem.

Subjects: Mathematics.


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