Answer
$2\left( \cos \frac{7\pi }{6}+i\sin \frac{7\pi }{6} \right)$
Work Step by Step
Division of two complex numbers:
For diving two complex numbers ${{z}_{1}}={{r}_{1}}\left( \cos {{\theta }_{1}}+i\sin {{\theta }_{1}} \right)$ and ${{z}_{2}}={{r}_{2}}\left( \cos {{\theta }_{2}}+i\sin {{\theta }_{2}} \right)$, we use the formula given below:
$\frac{{{z}_{1}}}{{{z}_{2}}}=\frac{{{r}_{1}}}{{{r}_{2}}}\left[ \cos \left( {{\theta }_{1}}-{{\theta }_{2}} \right)+i\sin \left( {{\theta }_{1}}-{{\theta }_{2}} \right) \right]$
So, the product of the given complex numbers ${{z}_{1}}=2\left( \cos \frac{5\pi }{3}+i\sin \frac{5\pi }{3} \right)\text{ and }{{z}_{2}}=\left( \cos \frac{\pi }{2}+i\sin \frac{\pi }{2} \right)$ can be given by:
$\begin{align}
& \frac{{{z}_{1}}}{{{z}_{2}}}=\frac{2}{1}\left[ \cos \left( \frac{5\pi }{3}-\frac{\pi }{2} \right)+i\sin \left( \frac{5\pi }{3}-\frac{\pi }{2} \right) \right] \\
& =2\left[ \cos \left( \frac{10\pi -3\pi }{6} \right)+i\sin \left( \frac{10\pi -3\pi }{6} \right) \right] \\
& =2\left( \cos \frac{7\pi }{6}+i\sin \frac{7\pi }{6} \right)
\end{align}$ Hence, the product of the complex numbers is $2\left( \cos \frac{7\pi }{6}+i\sin \frac{7\pi }{6} \right)$.
