Chapter 9.3, Problem 7CFU

To determine

To find: the rectangular coordinates of the point.

Answer to Problem 7CFU

The rectangular coordinates of point is $\left(-1,-\sqrt{3}\right)$ .

Explanation of Solution

Given information:

The polar coordinates are $\left(-2,\frac{4\pi }{3}\right)$ .

Calculation: since, the polar coordinates are $\left(-2,\frac{4\pi }{3}\right)$ .

To find the rectangular coordinates, substitute $x=r\cos \theta$ and $y=r\sin \theta$ ,

  $\begin{array}{l}x=r\cos \theta \\ x=-2\cos \left(\frac{\pi }{3}\right)\text{ [susbtitute]}\\ x=-2\left(\frac{1}{2}\right)\text{ [simplify]}\\ x=-1\\ y=r\sin \theta \\ y=-2\sin \left(\frac{\pi }{3}\right)\text{ [substitute]}\\ y=-2\left(\frac{\sqrt{3}}{2}\right)\text{ [simplify]}\\ y=-\sqrt{3}\end{array}$

Thus, the rectangular coordinates are $\left(-1,-\sqrt{3}\right)$ .