Chapter 9, Problem 2RE

To determine

To plot:the given points in the polar coordinates and find the rectangular coordinates.

Answer to Problem 2RE

  ${\left(4,\frac{2\pi }{3}\right)}_{r\theta }={\left(-2,2\sqrt{3}\right)}_{xy}$ .

Explanation of Solution

Given:

The givenpint is $\left(4,\frac{2\pi }{3}\right)$ .

Calculation:

As it is known that $x=r\cos \theta y=r\sin \theta$ .

And the given pint is $\left(4,\frac{2\pi }{3}\right)$ .

Therefore,

  $\begin{array}{l}x=4\cos \frac{2\pi }{3}=4\cos \left(\pi -\frac{\pi }{3}\right)=-4\cos \left(\frac{\pi }{3}\right)=-4\left(\frac{1}{2}\right)=-2\\ y=4\sin \frac{2\pi }{3}=4\sin \left(\pi -\frac{\pi }{3}\right)=4\sin \left(\frac{\pi }{3}\right)=4\left(\frac{\sqrt{3}}{2}\right)=2\sqrt{3}\\ \therefore {\left(4,\frac{2\pi }{3}\right)}_{r\theta }={\left(-2,2\sqrt{3}\right)}_{xy}\end{array}$

Hence, the rectangular coordinates of the given point is ${\left(4,\frac{2\pi }{3}\right)}_{r\theta }={\left(-2,2\sqrt{3}\right)}_{xy}$ .