Answer
The rectangular form of the given complex number is $4\sqrt{2}-4\sqrt{2}i$.
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 the 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),
$\begin{align}
& z=8\left( \cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4} \right) \\
& z=x+iy \\
\end{align}$
Simplify it further to get,
$\begin{align}
& 8\left( \cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4} \right)=8\cos \frac{7\pi }{4}+8i\sin \frac{7\pi }{4} \\
& =\left( 8\times \frac{1}{\sqrt{2}} \right)+i\left( 8\times -\frac{1}{\sqrt{2}} \right) \\
& =4\sqrt{2}-4\sqrt{2}i \\
& 8\left( \cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4} \right)=4\sqrt{2}-4\sqrt{2}i
\end{align}$
The rectangular form of the complex number is $4\sqrt{2}-4\sqrt{2}i$.
The rectangular form of the given complex number is $4\sqrt{2}-4\sqrt{2}i$.
