Question

In Problems $33-40$, find $zw$and $\frac{z}{w}$.Leave your answers in polar form.

$$\begin{gathered} z=\cos {120}^{\circ }+i\sin {120}^{\circ } \\ w=\cos {100}^{\circ }+i\sin {100}^{\circ } \end{gathered}$$

Short answer

The polar form of $zw$is, $\cos {220}^{\circ }+i\sin {220}^{\circ }$.

The polar form of$\frac{z}{w}$is,$\cos {20}^{\circ }+i\sin {20}^{\circ }$.

Step-by-step solution

Step 1 Given that,

$$\begin{gathered} z=\cos {120}^{\circ }+i\sin {120}^{\circ } \\ w=\cos {100}^{\circ }+i\sin {100}^{\circ } \end{gathered}$$

First find $zw$.

$$\begin{gathered} z.w=\left(\cos {120}^{\circ }+i\sin {120}^{\circ }\right).\left(\cos {100}^{\circ }+i\sin {100}^{\circ }\right) \\ =\cos \left({120}^{\circ }+{100}^{\circ }\right)+i\sin \left({120}^{\circ }+{100}^{\circ }\right) \\ =\cos {220}^{\circ }+i\sin {220}^{\circ } \end{gathered}$$

Step 2 Next find zw.

$$\begin{gathered} \frac{z}{w}=\frac{\cos {120}^{\circ }+i\sin {120}^{\circ }}{\cos {100}^{\circ }+i\sin {100}^{\circ }} \\ =\cos \left({120}^{\circ }-{100}^{\circ }\right)+i\sin \left({120}^{\circ }-{100}^{\circ }\right) \\ =\cos {20}^{\circ }+i\sin {20}^{\circ } \end{gathered}$$