Answer
The rectangular form of the given complex number is $-7i$.
Work Step by Step
Consider any complex number, given by $z=x+iy$; for a complex number in rectangular form,
$z=x+iy$ …… (1)
The polar form is given by,
$z=r\left( \cos \theta +i\sin \theta \right)$ …… (2)
Here, $x=r\cos \theta \ \text{ and }\ y=r\sin \theta $.
Divide the value of y by x, to get,
$\tan \theta =\frac{y}{x}$
Also, the value of r is called as moduli of the complex number, given by,
$r=\sqrt{{{x}^{2}}+{{y}^{2}}}$
For any complex number in polar form, $z=r\left( \cos \theta +i\sin \theta \right)$, the rectangular form is,
Using (1) and (2),
$\begin{align}
& z=7\left( \cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2} \right) \\
& z=x+iy \\
\end{align}$
Simplify it further to get,
$\begin{align}
& 7\left( \cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2} \right)=7\cos \frac{3\pi }{2}+7i\sin \frac{3\pi }{2} \\
& =\left( 7\times 0 \right)+i\left( 7\times -1 \right) \\
& =-7i \\
& 7\left( \cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2} \right)=-7i
\end{align}$
The rectangular form of the complex number is $-7i$
