Chapter 8.1, Problem 9E

To determine

To calculate: To plot the point that has the polar coordinates and then give two other polar coordinate representation of the point where $r<0$ and $r>0$

Answer to Problem 9E

Polar coordinates of point on the graph is plotted and two other polar coordinate representation of given points are $\left(3,\frac{5\pi }{2}\right),\left(-3,\frac{3\pi }{2}\right)$

Explanation of Solution

Given: Point is $\left(3,\frac{\pi }{2}\right)$

Formula Used:

Polar Coordinates are drawn with coordinates $\left(r,\theta \right)$ where $r=$ radius and $\theta =$ angle of the curve

First we need to convert Polar Coordinates to Cartesian Coordinates as follows:

  $\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Also, Point $\left(r,\theta \right)$ can be represented as

  $\left(r,\theta \right)=\left(r,\theta \pm 2n\pi \right)$

Or

  $\left(r,\theta \right)=\left(-r,\theta \pm \left(2n+1\right)\pi \right)$

Calculation:

  $\left(x,y\right)=\left(3cos\left(\pi /2\right),3\sin \left(\pi /2\right)\right)$

  $\begin{array}{l}=\left(3\left(0\right),3\left(1\right)\right)\\ =\left(0,3\right)\end{array}$

Plotting the polar coordinates on the graph

  

Now, let’s find two additional polar representations of the point, where $r<0$ and $r>0$

Point $\left(r,\theta \right)$ can be represented as

  $\left(r,\theta \right)=\left(r,\theta \pm 2n\pi \right)$

Or

  $\left(r,\theta \right)=\left(-r,\theta \pm \left(2n+1\right)\pi \right)$

Here,

  $\begin{array}{l}r=3\\ \theta =\frac{\pi }{2}\end{array}$

Substituting the values in above representations, additional representation of given point are:

  $\begin{array}{l}\left(3,\frac{\pi }{2}+2\pi \right)=\left(3,\frac{5\pi }{2}\right)\\ \left(-3,\frac{\pi }{2}+\pi \right)=\left(-3,\frac{3\pi }{2}\right)\end{array}$

Conclusion:

Hence, polar coordinates of point on the graph is plotted and two other polar coordinate representation of given points are $\left(3,\frac{5\pi }{2}\right),\left(-3,\frac{3\pi }{2}\right)$