Chapter 9, Problem 9RE

To determine

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

Answer to Problem 9RE

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

Explanation of Solution

Given:

The givenpoint is $\left(0,-2\right)$ .

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 $\left(0,-2\right)$ .

Therefore,

  $\begin{array}{l}P\left(0,-2\right)=\left(x,y\right)\\ x=0,y=-2\\ \therefore r=\sqrt{x^2+y^2}\\ =\sqrt{{\left(0\right)}^2+{\left(-2\right)}^2}=\sqrt{4}=2\end{array}$

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

  $\begin{array}{l}As,\tan \theta =\frac{y}{x}=\frac{-2}{0}=\infty \\ \therefore \theta =-\frac{\pi }{2}\end{array}$

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

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

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