Chapter 8.1, Problem 34E

To determine

To find the rectangular coordinates for the below polar coordinates -

  $\left(\sqrt{3},-5\pi /3\right)$

Answer to Problem 34E

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

Explanation of Solution

Given:

Polar coordinates - $\left(\sqrt{3},-5\pi /3\right)$ .

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 Polar coordinates are $\left(\sqrt{3},-5\pi /3\right)$

Using the above definition, we have:

  $r=\sqrt{3}$ and $\theta =-5\pi /3$

  $\begin{array}{l}x=r\cos \theta \\ x=\sqrt{3}\times \cos \left(-5\pi /3\right)\\ x=\sqrt{3}\times \cos \left(5\pi /3\right)\\ x=\sqrt{3}\times \frac{1}{2}\\ x=\frac{\sqrt{3}}{2}\end{array}$

And,

  $y=r\sin \theta$

  $\begin{array}{l}y=\sqrt{3}\times \sin \left(-5\pi /3\right)\\ y=-\sqrt{3}\times \sin \left(5\pi /3\right)\\ y=-\sqrt{3}\times \left(-\frac{\sqrt{3}}{2}\right)\\ y=\frac{3}{2}\end{array}$

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

Conclusion:

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