Chapter 9.5, Problem 10E

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 10E

The polar coordinates of point on the graph is plotted and three additional representation of given points are $\left(2,-\frac{5\pi }{4}\right),\left(-2,\frac{7\pi }{4}\right),\left(-2,-\frac{\pi }{4}\right)$

Explanation of Solution

Given information: Points is $\left(2,\frac{3\pi }{4}\right)$

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

  $\left(2,\frac{3\pi }{4}\right)$

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,

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

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

  $\begin{array}{l}\left(2,\frac{3\pi }{4}-2\pi \right)=\left(2,-\frac{5\pi }{4}\right)\\ \left(-2,\frac{3\pi }{4}+\pi \right)=\left(-2,\frac{7\pi }{4}\right)\\ \left(-2,\frac{3\pi }{4}-\pi \right)=\left(-2,-\frac{\pi }{4}\right)\end{array}$

Conclusion:

Hence, polar coordinates of point on the graph is plotted and three additional representation of given points are $\left(2,-\frac{5\pi }{4}\right),\left(-2,\frac{7\pi }{4}\right),\left(-2,-\frac{\pi }{4}\right)$