Question

In Problems 10–12, perform the given operation, where $z=2\left(\cos 85°+i\sin 85°\right)$and $w=3\left(\cos 22°+i\sin 22°\right)$. Write your answer in polar form.

$\frac{w}{z}$

Short answer

The answer is$\frac{w}{z}=\frac{3}{2}\left(\cos 297°+i\sin 297°\right)$

Step-by-step solution

Step 1. Given information

We have been given $z=2\left(\cos 85°+i\sin 85°\right)$and $w=3\left(\cos 22°+i\sin 22°\right)$.

We have to perform the operation$\frac{w}{z}$

Step 2. Perform the operation

$$\begin{gathered} \frac{w}{z} \\ =\frac{3\left(\cos 22°+i\sin 22°\right)}{2\left(\cos 85°+i\sin 85°\right)} \\ =\frac{3}{2}\left[\cos \left(22°-85°\right)+i\sin \left(22°-85°\right)\right] \\ =\frac{3}{2}\left(\cos \left(-63°\right)+i\sin \left(-63°\right)\right) \\ =\frac{3}{2}\left(\cos \left(360°-63°\right)+i\sin \left(360°-63°\right)\right) \\ =\frac{3}{2}\left(\cos 297°+i\sin 297°\right) \end{gathered}$$