Answer
See the explanation below.
Work Step by Step
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.
