Question

Find zw and write each answer in polar and exponential form.

Answer 1

Given the values of z and w, as

$\begin{gathered} z=9\mleft(cos\: \frac{\pi}{4}+i\: sin\: \frac{\pi}{4}\mright) \\ w=9\mleft(cos\: \frac{\pi\:}{10}+i\: sin\: \frac{\pi\:}{10}\mright) \end{gathered}$

we will find z w, as follow

$\begin{gathered} zw=9\lbrack(\cos \frac{\pi}{10}-\cos \frac{\pi}{4})+i(\sin \frac{\pi}{10}-\sin \frac{\pi}{4})\rbrack \\ zw=9\lbrack0.244-i0.398\rbrack \end{gathered}$

Simplifying further

$zw=2.196-i3.582$

To express in polar and exponetial form

$\begin{gathered} |r|=\sqrt[]{2.196^2+\mleft(-3.582\mright)^2} \\ |r|=4.20 \end{gathered}$

We need to get the angle between them

$\begin{gathered} \theta=\tan ^{-1}(\frac{-3.582}{2.196}) \\ \theta=-58.49^0 \\ \theta=301.511^0 \\ In\text{ radians} \\ \theta=\frac{2\pi\times301.511}{360}=1.675\pi \end{gathered}$

Thus, in the polar form, we will have

$(4.2,1.675\pi)$

For the exponential form, we will have

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