Answer
The multiplication of the complex numbers in the polar form is $30\left( \cos \frac{11\pi }{12}+i\sin \frac{11\pi }{12} \right)$.
Work Step by Step
Consider any complex number, given by,
$\begin{align}
& {{z}_{1}}={{r}_{1}}\left( \cos {{\theta }_{1}}+i\sin {{\theta }_{1}} \right) \\
& {{z}_{2}}={{r}_{2}}\left( \cos {{\theta }_{2}}+i\sin {{\theta }_{2}} \right) \\
\end{align}$
For a complex number in polar form, the multiplication is calculated as,
${{z}_{1}}\times {{z}_{2}}={{r}_{1}}{{r}_{2}}\left( \cos \left( {{\theta }_{1}}+{{\theta }_{2}} \right)+i\sin \left( {{\theta }_{1}}+{{\theta }_{2}} \right) \right)$ …… (1)
The polar form after the multiplication of the complex numbers,
$\begin{align}
& {{z}_{1}}=3\left( \cos \frac{5\pi }{8}+i\sin \frac{5\pi }{8} \right) \\
& {{z}_{2}}=10\left( \cos \frac{\pi }{16}+i\sin \frac{\pi }{16} \right) \\
\end{align}$
Multiply it using (1),
$\begin{align}
& {{z}_{1}}\times {{z}_{2}}={{r}_{1}}{{r}_{2}}\left( \cos \left( {{\theta }_{1}}+{{\theta }_{2}} \right)+i\sin \left( {{\theta }_{1}}+{{\theta }_{2}} \right) \right) \\
& {{z}_{1}}\times {{z}_{2}}=3\times 10\left( \cos \left( \frac{5\pi }{8}+\frac{\pi }{16} \right)+i\sin \left( \frac{5\pi }{8}+\frac{\pi }{16} \right) \right) \\
& =30\left( \cos \frac{11\pi }{12}+i\sin \frac{11\pi }{12} \right)
\end{align}$
Therefore,
The multiplication of the complex numbers in the polar form is $30\left( \cos \frac{11\pi }{12}+i\sin \frac{11\pi }{12} \right)$
