Chapter 9, Problem 6RE

To determine

To find: The two pairs of polar coordinates.

Answer to Problem 6RE

The two pairs of polar coordinates is $\left(-5,4.07\right)$ .

Explanation of Solution

Given information:

The given polar coordinates are $\left(3,4\right)$ .

Calculation:

Calculate the polar coordinates are $\left(3,4\right)$ .

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

Since, $x\ne 0$ and $y\ne 0$ substitute 3 for $x$ and 4 for $y$ in $r=\sqrt{x^2+y^2}$ and get the value of $r$ .

  $\begin{array}{c}r=\sqrt{3^2+4^2}\\ =\sqrt{9+16}\\ =5\end{array}$

Where $r=5$ the point $\left(3,4\right)$ lies in quadrant 1^st which means $0<\theta <\frac{\pi }{2}$ , where $\theta$ is the angle subtended by the ray with the polar axis.

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{4}{3}\right)\\ =0.93\end{array}$

But when $r=-5$ , the ray happens to be in the opposite direction of the terminal side of $\theta$ .

Now find the angle $\theta$ by substituting 3 for $x$ and 4 for $y$ in $\theta ={\tan }^{-1}\left(\frac{y}{x}\right),$

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{4}{3}\right)\\ ={\tan }^{-1}\left(1.33\right)\\ =4.07\end{array}$

Therefore, $r=-5$ , and $\theta =4.07$ ,we can write the polar coordinates of the given point as $\left(-5,4.07\right)$ .