Chapter 10.7, Problem 18E

To determine

Plot the point in the polar coordinates and find three additional polar representations.

Answer to Problem 18E

Additional polar representations are

  $\left(–5,5.49\right)$

  $\left(5,0.79\right)$

Explanation of Solution

Given:

  $\left(–5,–2.36\right)$

Consider the polar coordinate as: $\left(–5,–2.36\right)$

So, the directed distance from origin is:

  $r=–5$

Also, the directed angle is:

  $\theta =-2.36$

Convert this into pi radian as:

  $\begin{array}{cc}\theta & =-\frac{2.36}{3.14}\times \pi \\ & =-\frac{3\pi }{4}\end{array}$

Now, draw this polar coordinate in xy-plane as below. First consider the angle. So, as the angle is negative, so the count of angle will be clockwise. So, the considered angle will be as below:

Now,the directed distance is −5. So, for the negative sign the vector will be on the just opposite to the dotted line mentioned above. So, the vector will be as below:

Now, find the two additional polar representations of the points

So, consider the positive angle. But, the angle is clockwise. So, the directed angle will be:

  $\begin{array}{cc}\theta & =\pi +\frac{3\pi }{4}\\ & =\frac{4\pi +3\pi }{4}\\ & =\frac{7\pi }{4}\end{array}$

Convert into radian as:

  $\theta =\frac{7\pi }{4}\times \frac{3.14}{\pi }$

  $=5.49$

Now, angle is positive and the plot is clockwise. So, the value of r is negative. So, the directed distance is:

  $r=–5$

So, the polar coordinate is : $\left(–5,5.49\right)$

So, consider the positive angle. So, the angle will be anticlockwise. So, the directed angle will be:

  $\begin{array}{cc}\theta & =\pi -\frac{3\pi }{4}\\ & =\frac{\pi }{4}\end{array}$

Convert into radian as:

  $\theta =\frac{\pi }{4}\times \frac{3.14}{\pi }$

  $=0.79$

Now, angle is positive and the plot is anticlockwise. So, the value of r is positive. So, the directed distance is:

  $r=5$

So, the polar coordinate is: $\left(5,0.79\right)$