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\: (\pi)/(4)+i\: sin\: (\pi)/(4)\mright) \\ w=9\mleft(cos\: (\pi\:)/(10)+i\: sin\: (\pi\:)/(10)\mright) \end{gathered}`

we will find z w, as follow

`\begin{gathered} zw=9\lbrack(\cos (\pi)/(10)-\cos (\pi)/(4))+i(\sin (\pi)/(10)-\sin (\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)((-3.582)/(2.196)) \\ \theta=-58.49^0 \\ \theta=301.511^0 \\ In\text{ radians} \\ \theta=(2\pi*301.511)/(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