Chapter 11, Problem 17RE

To determine

To find: The x-coordinates of $3\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)$ .

Answer to Problem 17RE

The rectangular coordinates is $-\frac{3}{2}+\frac{3\sqrt{3}}{2}i$ .

Explanation of Solution

Given information:

The rectangular coordinates is $3\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)$ .

Calculation:

Calculate the x-coordinates,

If $3\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)$ represents a complex number in the polar form, then its $x$ and $y$ coordinates are $x=r\cos \theta$ and $y=r\sin \theta$ , where $\left(r,\theta \right)$ .

The x-coordinates of $3\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)$ is,

  $\begin{array}{c}x=r\cos \theta \\ =3\cos \left(\frac{2\pi }{3}\right)\\ =3\left(-\frac{1}{2}\right)\\ =-\frac{3}{2}\end{array}$

The y-coordinates is

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

  $\begin{array}{c}z=x+iy\\ =-\frac{3}{2}+\frac{3\sqrt{3}}{2}i\end{array}$

Therefore, the rectangular form of $3\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)$ is $-\frac{3}{2}+\frac{3\sqrt{3}}{2}i$

The rectangular coordinates of the given point $z=-\frac{3}{2}+\frac{3\sqrt{3}}{2}i$ are $\left(-\frac{3}{2}+\frac{3\sqrt{3}}{2}i\right)$

The point is plotted in the complex plane figure-(1) shown .

  

Figure-(1)

Therefore, the rectangular coordinates is $-\frac{3}{2}+\frac{3\sqrt{3}}{2}i$ .