Answer
The rectangular form of the complex number is $-2\sqrt{3}+2i$.
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 $, dividing 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),
$\begin{align}
& z=4\left( \cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6} \right) \\
& z=x+iy \\
\end{align}$
Simplify it further to get,
$\begin{align}
& 4\left( \cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6} \right)=4\cos \frac{5\pi }{6}+4i\sin \frac{5\pi }{6} \\
& =\left( 4\times -\frac{\sqrt{3}}{2} \right)+i\left( 8\times -\frac{1}{\sqrt{2}} \right) \\
& =-2\sqrt{3}+2i \\
& 4\left( \cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6} \right)=-2\sqrt{3}+2i
\end{align}$
The rectangular form of the complex number is $-2\sqrt{3}+2i$.
Hence, the rectangular form of the complex number is $-2\sqrt{3}+2i$.
