Answer
The rectangular form of $z=4\left( \cos 210{}^\circ +i\sin 210{}^\circ \right)$ is $z=\left( -2\sqrt{3}-2i \right)$.
Work Step by Step
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,
$\begin{align}
& z=4\left( \cos 210{}^\circ +i\sin 210{}^\circ \right) \\
& =4\left( -\frac{\sqrt{3}}{2}-i\frac{1}{2} \right) \\
& =\left( -\frac{4\sqrt{3}}{2}-i\frac{4}{2} \right) \\
& =\left( -2\sqrt{3}-2i \right)
\end{align}$
