Chapter 8.1, Problem 23E

To determine

To calculate: To find the polar coordinates for the rectangular point with $r>0$ and $0<\theta <2\pi$

Answer to Problem 23E

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

Explanation of Solution

Given: Rectangular point on graph 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 tha rectangular point on graph as follows:

  

Here rectangular coordinates are $P\left(-3,3\right)$

Using the above definition,:

  $x=-3$ and $y=3$

  $\begin{array}{l}r=\sqrt{{\left(-3\right)}^2+{\left(3\right)}^2}\\ r=\sqrt{9+9}\\ r=\sqrt{18}\\ r=\pm 3\sqrt{2}\end{array}$

Since, $r>0$ , therefore $r=3\sqrt{2}$

Now,

  $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

  $\begin{array}{l}\theta ={\tan }^{-1}\left(\frac{3}{-3}\right)\\ \theta ={\tan }^{-1}\left(-1\right)\\ \theta =-{\tan }^{-1}\left(1\right)\\ \theta =-\frac{\pi }{4}\end{array}$

Since, $0<\theta <2\pi$ , therefore, $\theta =\frac{3\pi }{4}$

Thus, the polar coordinates are $\left(3\sqrt{2},\frac{3\pi }{4}\right)$ .

Conclusion:

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