Question

Find $zw$and $\frac{z}{w}$. Leave your answers in polar form.

$$\begin{gathered} z=3\left(\cos \frac{9\pi }{5}+i\sin \frac{9\pi }{5}\right) \\ w=2\left(\cos \frac{\pi }{5}+i\sin \frac{\pi }{5}\right) \end{gathered}$$

Short answer

We get:

$$\begin{gathered} \mathrm{z}\cdot \mathrm{w}=6[cos2\pi +\mathrm{isin}2\pi ] \\ \frac{\mathrm{z}}{\mathrm{w}}=\frac{3}{2}\cdot \left(\cos \frac{8\mathrm{\pi }}{5}+\mathrm{isin}\frac{8\mathrm{\pi }}{5}\right) \end{gathered}$$

Step-by-step solution

Step 1. Given information

We have been given:

$$\begin{gathered} z=3\left(\cos \frac{9\pi }{5}+i\sin \frac{9\pi }{5}\right) \\ w=2\left(\cos \frac{\pi }{5}+i\sin \frac{\pi }{5}\right) \end{gathered}$$

We have to find$zw$ and $\frac{z}{w}$.

Step 2. Write the formula for the multiplication division of the complex numbers 

$\begin{array}{r}{Z}_1{Z}_2=r_1{r}_2\left[\cos \left({\theta }_1+{\theta }_2\right)+i\sin \left({\theta }_1+{\theta }_2\right)\right]\\ \frac{Z_1}{Z_2}=\frac{r_1}{r_2}\left[\cos \left({\theta }_1-{\theta }_2\right)+i\sin \left({\theta }_1-{\theta }_2\right)\right]\end{array}$

Step 3. Find zw.  

$$\begin{gathered} \mathrm{z}\cdot \mathrm{w}=3\left(cos\frac{9\pi }{5}+\mathrm{isin}\frac{9\pi }{5}\right)\cdot 2\left(cos\frac{\pi }{5}+\mathrm{isin}\frac{\pi }{5}\right) \\ \mathrm{z}\cdot \mathrm{w}=6\left(cos(\frac{{\displaystyle 9\pi }}{{\displaystyle 5}}+\frac{{\displaystyle \pi }}{{\displaystyle 5}})+\mathrm{isin}(\frac{{\displaystyle 9\pi }}{{\displaystyle 5}}+\frac{{\displaystyle \mathrm{\pi }}}{{\displaystyle 5}})\right) \\ \mathrm{z}\cdot \mathrm{w}=6\left(cos2\mathrm{\pi }+\mathrm{isin}2\mathrm{\pi }\right) \end{gathered}$$

Step 4. Find zw.

$$\begin{gathered} \frac{z}{w}=\frac{3\left(cos\frac{9\pi }{5}+\mathrm{isin}\frac{9\pi }{5}\right)}{2\left(cos\frac{\pi }{5}+\mathrm{isin}\frac{\pi }{5}\right)} \\ =\frac{3}{2}\cdot \left[\cos \left(\frac{9\pi }{5}-\frac{\pi }{5}\right)+\mathbf{i}\sin \left(\frac{9\pi }{5}-\frac{\pi }{5}\right)\right] \\ =\frac{3}{2}\cdot \left(\cos \frac{8\pi }{5}+\mathrm{isin}\frac{8\pi }{5}\right) \end{gathered}$$