Chapter 8.1, Problem 12E

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 12E

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

Explanation of Solution

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

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:

Point is given as

  $\left(-2,-\frac{\pi }{3}\right)$

  $\begin{array}{l}\left(x,y\right)=\left(-2cos\left(-\frac{\pi }{3}\right),-1\sin \left(-\frac{\pi }{3}\right)\right)\\ \left(x,y\right)=\left(-2\left(\frac{1}{2}\right),-1\left(-\frac{\sqrt{3}}{2}\right)\right)\\ \left(x,y\right)=\left(-1,1.73\right)\end{array}$

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,

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

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

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

Conclusion:

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