Chapter 8.1, Problem 25E

To determine

To calculate: To find the rectangular coordinates for the polar form -

Answer to Problem 25E

The rectangular coordinates are $\left(-2,-2\sqrt{3}\right)$ .

Explanation of Solution

Given: Polar coordinate is

  

Formula Used:

A polar equation is any equation that describes a relation between r and θ, where r represents the distance from the pole (origin) to a point on a curve, and θ represents the counter-clockwise angle made by a point on a curve, the pole, and the positive x-axis.

Also, $x=r\cos \theta$ and $y=r\sin \theta$

  $r=\sqrt{x^2+y^2}$ and $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

Calculation:

Given the polar coordinate of point as follows:

  

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

Using the above definition:

  $r=4,\theta =-\frac{2\pi }{3}$

  $\begin{array}{l}x=r\cos \theta \\ x=4\cos \left(-\frac{2\pi }{3}\right)\\ x=4\times \cos \left(\frac{2\pi }{3}\right)\\ x=-4\times \frac{1}{2}\\ x=-2\end{array}$

Now,

  $\begin{array}{l}y=r\sin \theta \\ y=4\times \sin \left(-\frac{2\pi }{3}\right)\\ y=-4\times \sin \left(\frac{2\pi }{3}\right)\\ y=-4\times \frac{\sqrt{3}}{2}\\ y=-2\sqrt{3}\end{array}$

Thus, the rectangular coordinates are $\left(-2,-2\sqrt{3}\right)$ .

Conclusion:

The rectangular coordinates are $\left(-2,-2\sqrt{3}\right)$ .