Chapter 9, Problem 7RE

To determine

To find:the rectangular coordinates of the given point.And also find two pairs of polar coordinates.

Answer to Problem 7RE

  $P\left(3\sqrt{2},\frac{3\pi }{4}\right),P\left(-3\sqrt{2},-\frac{\pi }{4}\right)$ .

Explanation of Solution

Given:

The givenpoint is $\left(-3,3\right)$ .

Calculation:

According to the question, it is to find two pairs of polar coordinates for ach point, one with $r>0$ and other with $r<0$ and express $\theta$ in radians.

And the given point is $\left(-3,3\right)$ .

Therefore,

  $\begin{array}{l}P\left(-3,3\right)=\left(x,y\right)\\ x=-3,y=3\\ \therefore r=\sqrt{x^2+y^2}\\ =\sqrt{-3^2+3^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}\end{array}$

Now compute the value of $\tan \theta$ so that to find $\theta$ .

  $\begin{array}{l}As,\tan \theta =\frac{y}{x}=\frac{3}{-3}=-1\\ \therefore \theta ={\tan }^{-1}\left(-1\right)\\ \Rightarrow \theta =\pi -\frac{\pi }{4}=\frac{3\pi }{4}\end{array}$

Now, to find pair of polar coordinates with $r>0$ and other with $r<0$ .

  $\begin{array}{l}P\left(3\sqrt{2},\frac{3\pi }{4}\right)\left\{\text{as},r>0\right\}\\ P\left(-3\sqrt{2},-\frac{\pi }{4}\right)\left\{\text{as},r<0\right\}\end{array}$

Hence, the rectangular coordinates of the given point is $P\left(3\sqrt{2},\frac{3\pi }{4}\right),P\left(-3\sqrt{2},-\frac{\pi }{4}\right)$ .