Answer
The rectangular form of $z=8\left( \cos 60{}^\circ +i\sin 60{}^\circ \right)$ is $z=4+4i\sqrt{3}$.
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 $r=8$ and $\theta =60{}^\circ $.
Since, $\cos 60{}^\circ =\frac{1}{2}\ \ \text{ and }\ \sin 60{}^\circ =\frac{\sqrt{3}}{2}\ \ $
Therefore,
$\begin{align}
& z=8\left( \cos 60{}^\circ +i\sin 60{}^\circ \right) \\
& =8\left( \frac{1}{2}+i\frac{\sqrt{3}}{2} \right) \\
& =\left( \frac{8}{2}+i\frac{\sqrt{3}\times 8}{2} \right) \\
& =\left( 4+4i\sqrt{3} \right)
\end{align}$
