## Quick Reference

From the definition of multiplication (of a complex number), it follows that (cos *θ*_{1}+*i* sin *θ*_{1})(cos *θ*_{2}+*i* sin *θ*_{2})=cos (*θ*_{1}+*θ*_{2})+*i* sin (*θ*_{1}+*θ*_{2}). This leads to the following result known as De Moivre's Theorem, which is crucial to any consideration of the powers *z*^{n} of a complex number *z*:

Theorem

For all positive integers *n*, (cos *θ*+*i* sin *θ*)^{n}=cos *nθ*+*i* sin *nθ*.

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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