Chapter 9, Problem 4RE

To determine

To find: The two pairs of polar coordinates.

Answer to Problem 4RE

The two pairs of polar coordinates is $\left(-3\sqrt{2},-\frac{\pi }{4}\right)$ .

Explanation of Solution

Given information:

The given polar coordinates $\left(-3,3\right)$ .

Calculation:

Calculate the polar coordinates is $\left(-3,3\right)$ .

Now calculate the distance $r$ of the point from the origin.

Since, $x\ne 0$ and $y\ne 0$ substitute -3 for $x$ and 3for $y$ in $r=\sqrt{x^2+y^2}$ and get the value of $r$ .

  $\begin{array}{c}r=\sqrt{{\left(-3\right)}^2+3^2}\\ =\sqrt{81}\\ =3\sqrt{2}\end{array}$

Where $r=3\sqrt{2}$ the point $\left(-3,3\right)$ lies in quadrant 2^nd which means $\frac{\pi }{2}<\theta <\pi$ , where $\theta$ is the angle subtended by the ray with the polar axis.

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

Then, $r=3\sqrt{2}$ and $\theta =\frac{3\pi }{4}$ ,we can write the polar coordinates of the given point as $\left(3\sqrt{2},\frac{3\pi }{4}\right)$ .

Since, the points $\left(-3,3\right)$ lies in quadrant 2^nd or $\frac{\pi }{2}<\theta <\pi$ , where $\theta$ in the angle subtended by the ray with the polar axis.

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

Therefore, $r=3\sqrt{2}$ , and $\theta =-\frac{\pi }{4}$ ,we can write the polar coordinates of the given point as $\left(-3\sqrt{2},-\frac{\pi }{4}\right)$ .