Chapter 8.1, Problem 10E

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 10E

Polar coordinates of point on the graph is plotted and two other polar coordinate representation of given points are $\left(2,\frac{11\pi }{4}\right),\left(2,\frac{7\pi }{4}\right)$

Explanation of Solution

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

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:

  $\begin{array}{l}\left(x,y\right)=\left(2cos\left(\frac{3\pi }{4}\right),2\sin \left(\frac{3\pi }{4}\right)\right)\\ \left(x,y\right)=\left(2\left(-\frac{1}{\sqrt{2}}\right),2\left(\frac{1}{\sqrt{2}}\right)\right)\\ \left(x,y\right)=\left(-\sqrt{2},\sqrt{2}\right)\end{array}$

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,

  $\begin{array}{l}r=2\\ \theta =\frac{3\pi }{4}\end{array}$

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

  $\begin{array}{l}\left(2,\frac{3\pi }{4}+2\pi \right)=\left(2,\frac{11\pi }{4}\right)\\ \left(-2,\frac{3\pi }{4}+\pi \right)=\left(2,\frac{7\pi }{4}\right)\end{array}$

Conclusion:

Hence, polar coordinates of point on the graph is plotted and two other polar coordinate representation of given points are $\left(2,\frac{11\pi }{4}\right),\left(2,\frac{7\pi }{4}\right)$