Answer
$\begin{align}
& \left[ 1\left( \cos \,90{}^\circ +i\,\sin \,90{}^\circ \right) \right]\left[ 2\sqrt{2}\left( \cos \,45{}^\circ +i\,\sin \,45{}^\circ \right) \right]\left[ 2\left( \cos \,150{}^\circ +i\,\sin \,150{}^\circ \right) \right]; \\
& 4\sqrt{2}\left( \cos \,285{}^\circ +i\,\sin \,285{}^\circ \right);\ \approx 1.4641-5.4641i \\
\end{align}$
Work Step by Step
Consider the provided expression
$z=i\left( 2+2i \right)\left( -\sqrt{3}+i \right)$
First convert it into polar form:
$\begin{align}
& z=i\left( 2+2i \right)\left( -\sqrt{3}+i \right) \\
& =\left[ 1\left( \cos \,90{}^\circ +i\,\sin \,90{}^\circ \right) \right]\left[ 2\sqrt{2}\left( \cos \,45{}^\circ +i\,\sin \,45{}^\circ \right) \right]\left[ 2\left( \cos \,150{}^\circ +i\,\sin \,150{}^\circ \right) \right]
\end{align}$
Apply the multiplication rule of complex numbers:
$\begin{align}
& z=\left( 1\times 2\sqrt{2}\times 2 \right)\left( \cos \left( 90{}^\circ +45{}^\circ +150{}^\circ \right)+i\,\sin \left( 90{}^\circ +45{}^\circ +150{}^\circ \right) \right) \\
& =4\sqrt{2}\left( \cos \,285{}^\circ +i\,\sin \,285{}^\circ \right)
\end{align}$
The above expression is the polar form of the provided expression.
Convert it into rectangular form:
$z=4\sqrt{2}\left( \cos \,285{}^\circ +i\,\sin \,285{}^\circ \right)$
Substitute the values of $\cos \,285{}^\circ $ and $\sin \,285{}^\circ $ in the above expression:
$\begin{align}
& z=4\sqrt{2}\left( 0.2588-i0.9659 \right) \\
& =1.4641-i5.4641
\end{align}$
The above expression is the rectangular form of the provided expression is:
The polar form of the provided expression is $4\sqrt{2}\left( \cos \,285{}^\circ +i\,\sin \,285{}^\circ \right)$.
The rectangular form of the provided expression is $\approx 1.4641-i5.4641$.
