Chapter 9, Problem 15RE

To determine

To find: The complex number in polar form.

Answer to Problem 15RE

The complex number in polar form of $z=4-3i$ is $5\left(\cos 323.1°+i\sin 323.1°\right)$ .

Explanation of Solution

Given information:

The given polar equation is $\left(4-3i\right)$ .

Calculation:

Calculate the rectangular coordinate is $z=4-3i$ .

The graph is plotted in the complex plane figure-(1) shows.

  

Figure-(1)

From the graph it is clear that the point is located in quadrant 4^th.

the distance $r$ of the complex number $z=x+iy$ from the origin to the point $\left(x,y\right)$ is

  $\begin{array}{c}\left|z\right|=r\\ r=\sqrt{x^2+y^2}\\ r=\sqrt{4^2+{\left(-3\right)}^2}\\ r=\sqrt{25}\\ r=5\end{array}$

Use $x=r\cos \theta$ and $y=r\sin \theta$ to find the value of $\theta$ .

  $\sin \theta =\frac{-3}{5}$ , $\cos \theta =\frac{4}{5}$ .

Since, $\sin 323.1°=\frac{-3}{5}$ , $\cos 323.1°=\frac{4}{5}$ , $\theta =323.1°$ .

Therefore, the polar form of $z=4-3i$ is $5\left(\cos 323.1°+i\sin 323.1°\right)$ .