## Precalculus (6th Edition) Blitzer

The provided complex number $z=r\left( \cos \theta +i\sin \theta \right)$ is in polar form with $r$ and $\theta$ as its modulus and argument respectively. First find the real and imaginary parts of the complex number, which are represented by $a$ and $b$ respectively. $a=r\cos \theta$ and $b=r\sin \theta$ Substitute the value of $r\cos \theta$ and $r\sin \theta$ in the provided complex number. \begin{align} & z=a+ib \\ & =\left( r\cos \theta \right)+i\left( r\sin \theta \right) \end{align} Above is the rectangular form of the assumed complex number. Example: The above explanation can be justified with the help of an example. Let the provided complex number be $z=5\left( \cos 30{}^\circ +i\sin 30{}^\circ \right)$, in polar form. To convert this complex number into rectangular form, just substitute the value of $\cos 30{}^\circ$ and $\sin 30{}^\circ$ in the provided expression. $z=5\left( \frac{\sqrt{3}}{2}+i\frac{1}{2} \right)$ On further simplification $z=\left( \frac{5\sqrt{3}}{2}+\frac{5}{2}i \right)$ The above expression is the rectangular form of the provided expression.