From the definition of multiplication (of a complex number), it follows that $\left(\text{c}\text{o}\text{s}{\theta }_1\text{+}i\text{s}\text{i}\text{n}{\theta }_1\right)\left(\text{c}\text{o}\text{s}{\theta }_2+i\text{s}\text{i}\text{n}{\theta }_2\right)=\text{c}\text{o}\text{s}\left({\theta }_1+{\theta }_2\right)+i\text{s}\text{i}\text{n}\left({\theta }_1+{\theta }_2\right).$ This leads to the following result known as De Moivre’s Theorem, which is crucial to calculating the powers zⁿ of a complex number z:

Theorem

For any integer n, ${\left(\text{c}\text{o}\text{s}\theta +i\text{s}\text{i}\text{n}\theta \right)}^n=\text{c}\text{o}\text{s}n\theta +i\text{s}\text{i}\text{n}n\theta .$