Chapter 6, Problem 21CT

To determine

The trigonometric form of the complexnumber.

Answer to Problem 21CT

The trigonometric form of the complex number is $z=4\sqrt{2}\left(\cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4}\right)$

Explanation of Solution

Given info:

  $z=4-4i$

Formula used:

The trigonometric form of the complex number $z=a+ib$ is

  $z=r\left(\cos \theta +i\sin \theta \right)$

Where $r=\sqrt{a^2+b^2}$ and $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$

Calculation:

We have

  $\begin{array}{l}z=4-4i\\ a=4,b=-4\\ r=\sqrt{a^2+b^2}=\sqrt{4^2+{\left(-4\right)}^2}=4\sqrt{2}\\ \theta ={\tan }^{-1}\left(\frac{-4}{4}\right)=\frac{7\pi }{4}\\ z=r\left(\cos \theta +i\sin \theta \right)\\ z=4\sqrt{2}\left(\cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4}\right)\end{array}$

The trigonometric form of the complex number is $z=4\sqrt{2}\left(\cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4}\right)$

Conclusion:

Thus,thetrigonometric form of the complex number is $z=4\sqrt{2}\left(\cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4}\right)$