Chapter 9, Problem 4CT

To determine

The polar coordinates $\left(r,\theta \right)$ of a rectangular coordinate $\left(2,2\sqrt{3}\right)$ .

Answer to Problem 4CT

Solution:

The polar coordinates of a rectangular coordinate $\left(2,2\sqrt{3}\right)$ is $\left(4,\frac{\pi }{3}\right)$ .

Explanation of Solution

Given information:

The rectangular coordinate, $\left(2,2\sqrt{3}\right)$

Step $1:$ Plot the point $\left(x,y\right)$ and note the quadrant the point lies in or the coordinate axis the

point lies on.

The point $\left(2,2\sqrt{3}\right)$ lies in the quadrant $\text{I}$ and is plotted in the rectangular coordinate system as shown below:

  

So, the $x-$ coordinate and the $y-$ coordinate both are positive.

Step $2:$ Determine the distance $r$ from the origin to the point using $r=\sqrt{x^2+y^2}$ .

By substituting the values of $x=2$ and $y=2\sqrt{3}$ in the above formula,

  $r=\sqrt{x^2+y^2}$

  $\Rightarrow r=\sqrt{{\left(2\right)}^2+{\left(2\sqrt{3}\right)}^2}=\sqrt{4+4\left(3\right)}=\sqrt{16}=4$

  $\Rightarrow r=4$

Step $3:$ Determine $\theta$ .

  $\tan \theta =\frac{y}{x}$

So, $\theta ={\tan }^{-1}\left(\frac{2\sqrt{3}}{2}\right)={\tan }^{-1}\left(\sqrt{3}\right)=\frac{\pi }{3}$

Therefore, the polar coordinates for the rectangular coordinate $\left(2,2\sqrt{3}\right)$ is $\left(4,\frac{\pi }{3}\right)$ .