Answer
The rectangular form of $z=6\left( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} \right)$ is $z=\left( -3+3i\sqrt{3} \right)$.
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 given complex number is in the polar form with $r=6$ and $\theta =\frac{2\pi }{3}$.
Since, $\cos \frac{2\pi }{3}=-\frac{1}{2}\ \text{ and }\ \sin \frac{2\pi }{3}=\frac{\sqrt{3}}{2}\ $
Therefore,
$\begin{align}
& z=6\left( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} \right) \\
& =6\left( -\frac{1}{2}+i\frac{\sqrt{3}}{2} \right) \\
& =\left( -\frac{6}{2}+i\frac{6\sqrt{3}}{2} \right) \\
& =\left( -3+3i\sqrt{3} \right)
\end{align}$
