Answer
$\frac{1}{2}\left( \cos \pi +i\sin \pi \right)$
Work Step by Step
Division of two complex numbers:
The division of two complex numbers ${{z}_{1}}={{r}_{1}}\left( \cos {{\theta }_{1}}+i\sin {{\theta }_{1}} \right)$ and ${{z}_{2}}={{r}_{2}}\left( \cos {{\theta }_{2}}+i\sin {{\theta }_{2}} \right)$, is given by
$\frac{{{z}_{1}}}{{{z}_{2}}}=\frac{{{r}_{1}}}{{{r}_{2}}}\left[ \cos \left( {{\theta }_{1}}-{{\theta }_{2}} \right)+i\sin \left( {{\theta }_{1}}-{{\theta }_{2}} \right) \right]$
So, the product of the given complex numbers ${{z}_{1}}=5\left( \cos \frac{4\pi }{3}+i\sin \frac{4\pi }{3} \right)\text{ and }{{z}_{2}}=10\left( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right)$ is
$\begin{align}
& \frac{{{z}_{1}}}{{{z}_{2}}}=\frac{5}{10}\left[ \cos \left( \frac{4\pi }{3}-\frac{\pi }{3} \right)+i\sin \left( \frac{4\pi }{3}-\frac{\pi }{3} \right) \right] \\
& =\frac{1}{2}\left( \cos \pi +i\sin \pi \right)
\end{align}$
Hence, the product of the complex numbers is $\frac{1}{2}\left( \cos \pi +i\sin \pi \right)$.
