## Precalculus (6th Edition) Blitzer

The rectangular form of $z=4\left( \cos 210{}^\circ +i\sin 210{}^\circ \right)$ is $z=\left( -2\sqrt{3}-2i \right)$.
The rectangular form of a complex number is $z=a+ib$, where $a$ and $b$ are rectangular coordinates. In the polar form the complex number $z=a+ib$ is represented as $z=r\left( \cos \theta +i\sin \theta \right)$, where r is the distance of the complex number from the origin and $\theta$ is the respective angle. Here, the provided complex number is in the polar form with $r=4$ and $\theta =210{}^\circ$. Since, $\cos 210{}^\circ =-\frac{\sqrt{3}}{2}\ \ \text{ and }\ \sin 210{}^\circ =-\frac{1}{2}$ Therefore, \begin{align} & z=4\left( \cos 210{}^\circ +i\sin 210{}^\circ \right) \\ & =4\left( -\frac{\sqrt{3}}{2}-i\frac{1}{2} \right) \\ & =\left( -\frac{4\sqrt{3}}{2}-i\frac{4}{2} \right) \\ & =\left( -2\sqrt{3}-2i \right) \end{align}