Chapter 9.5, Problem 11E

To determine

To calculate: To plot the point in polar coordinates and find three additional polar representations of the point, using $-2\pi <\theta <2\pi$

Answer to Problem 11E

The polar coordinates of point on the graph is plotted and three additional representation of given points are $\left(-1,\frac{2\pi }{3}\right),\left(-1,-\frac{4\pi }{3}\right)$

Explanation of Solution

Given information: Points is $\left(1,-\frac{\pi }{3}\right)$

Formula Used:

Polar coordinate is given as $\left(r,\theta \right)$ , where $r$ is radius of circle

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

  $\left(1,-\frac{\pi }{3}\right)$

Plotting the polar coordinates on the graph

  

Now, let’s find three additional polar representations of the point, using $-2\pi <\theta <2\pi$

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,

  $\begin{array}{l}r=1\\ \theta =-\frac{\pi }{3}\end{array}$

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

  $\begin{array}{l}\left(-1,-\frac{\pi }{3}+\pi \right)=\left(-1,\frac{2\pi }{3}\right)\\ \left(-1,-\frac{\pi }{3}-\pi \right)=\left(-1,-\frac{4\pi }{3}\right)\end{array}$

Conclusion:

Hence, polar coordinates of point on the graph is plotted and three additional representation of given points are $\left(-1,\frac{2\pi }{3}\right),\left(-1,-\frac{4\pi }{3}\right)$