## Precalculus (6th Edition) Blitzer

The multiplication of the complex numbers in the polar form is $30\left( \cos \frac{11\pi }{12}+i\sin \frac{11\pi }{12} \right)$.
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 $30\left( \cos \frac{11\pi }{12}+i\sin \frac{11\pi }{12} \right)$