Chapter 8.1, Problem 11E

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 11E

Polar coordinates of point on the graph is plotted and two other polar coordinate representation of given points are $\left(-1,\frac{19\pi }{6}\right),\left(1,\frac{13\pi }{6}\right)$

Explanation of Solution

Given: Point is $\left(-1,\frac{7\pi }{6}\right)$

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

  $\left(-1,\frac{7\pi }{6}\right)$

  $\begin{array}{l}\left(x,y\right)=\left(-1cos\left(\frac{7\pi }{6}\right),-1\sin \left(\frac{7\pi }{6}\right)\right)\\ \left(x,y\right)=\left(-1\left(-\frac{\sqrt{3}}{2}\right),-1\left(\frac{1}{2}\right)\right)\\ \left(x,y\right)=\left(0.86,-0.5\right)\end{array}$

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,

  $\begin{array}{l}r=-1\\ \theta =\frac{7\pi }{6}\end{array}$

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

  $\begin{array}{l}\left(-1,\frac{7\pi }{6}+2\pi \right)=\left(-1,\frac{19\pi }{6}\right)\\ \left(1,\frac{7\pi }{6}+\pi \right)=\left(1,\frac{13\pi }{6}\right)\end{array}$

Conclusion:

Hence, polar coordinates of point on the graph is plotted and two other polar coordinate representation of given points are $\left(-1,\frac{19\pi }{6}\right),\left(1,\frac{13\pi }{6}\right)$