Chapter 9.5, Problem 8E

To determine

To calculate: To match the polar coordinates with the point on the graph and then find the rectangular coordinates of the point

Answer to Problem 8E

The polar coordinates of point on the graph matches with point B and rectangular coordinate is $\left(\sqrt{2},\sqrt{2}\right)$

Explanation of Solution

Given information: Points is $\left(-2,\frac{5\pi }{4}\right)$

Polar coordinates of points

  

Formula Used:

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

Rectangular coordinate of the point in polar coordinate $\left(r,\theta \right)$ is given as

  $x=r\cos \theta$

  $y=r\sin \theta$

Calculation:

Point is given as

  $\left(-2,\frac{5\pi }{4}\right)$

Plotting the polar coordinates on the graph

  

Thus, the point on graph is B

Rectangular coordinate of the point is calculated as follows:

  $x=r\cos \theta$ and

  $y=r\sin \theta$

Calculating x coordinate,

  $\begin{array}{l}x=-2\cos \frac{5\pi }{4}\\ x=-2\times -\frac{1}{\sqrt{2}}\\ x=\sqrt{2}\end{array}$

Now, calculating y coordinate

  $\begin{array}{l}y=-2\sin \frac{5\pi }{4}\\ y=-2\times -\frac{1}{\sqrt{2}}\\ y=\sqrt{2}\end{array}$

Thus, rectangular coordinate is $\left(\sqrt{2},\sqrt{2}\right)$

Conclusion:

Hence, polar coordinates of point on the graph matches with point B and rectangular coordinate is $\left(\sqrt{2},\sqrt{2}\right)$