Chapter 9.5, Problem 19E

To determine

To calculate: To plot the point in the polar coordinates and find the rectangular coordinates of the point

Answer to Problem 19E

Point is plotted in the polar coordinates and rectangular coordinate of point is $\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$

Explanation of Solution

Given information: Points is $\left(1,-\frac{3\pi }{4}\right)$

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

  $\left(1,-\frac{3\pi }{4}\right)$

Plotting the polar coordinates on the graph

  

Rectangular coordinate of the point is calculated as follows:

  $x=r\cos \theta$ and

  $y=r\sin \theta$

Calculating x coordinate,

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

Now, calculating y coordinate

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

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

Conclusion:

Hence, point is plotted in polar coordinates and rectangular coordinate is $\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$