Chapter 9, Problem 16RE

To determine

To find: The complex number in standard form $a+bi$ .

Answer to Problem 16RE

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 $2\left(\cos 150°+i\sin 150°\right)$ .

Calculation:

Calculate the x-coordinate of $2\left(\cos 150°+i\sin 150°\right)$ is,

  $\begin{array}{c}x=r\cos \theta \\ =2\cos 150°\\ =2\left(-\frac{\sqrt{3}}{2}\right)\\ =-\sqrt{3}\end{array}$

The y-coordinate is

  $\begin{array}{c}y=r\sin \theta \\ =2\sin 150°\\ =2\left(\frac{\sqrt{3}}{2}\right)\\ =1\end{array}$

  $\begin{array}{c}z=x+iy\\ =-\sqrt{3}+i\end{array}$

Therefore, the rectangular form is $2\left(\cos 150°+i\sin 150°\right)$ is $-\sqrt{3}+i$ .

The rectangular coordinates of the given is point $z=-\sqrt{3}+i$ .

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 2^nd.

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