Chapter H.1, Problem 3E

(a)

To determine

To plot: the point and find the Cartesian coordinates of the point.

Answer to Problem 3E

Hence, the required Cartesian coordinate point is [(-1,0)]

Explanation of Solution

Given:

Polar coordinates are : $(1,\pi )$

Calculation:

Consider the following point : $(1,\pi )$

Draw the following plot

  

Compare the point $(1,\pi )$ with the polar point $(r,\theta )$

Here, $r=1,\theta =\pi$

Recollect that:

  $x=r\cos \theta ,y=r\sin \theta x=(1)\cos (\pi )=\cos \pi =-1y=(1)\sin (\pi )$

  $=\sin \pi$

  $=0$

Hence, the required Cartesian coordinate point is [(-1,0)]

Conclusion:

Hence, the required Cartesian coordinate point is [(-1,0)]

(b)

To determine

To plot: the point and find the Cartesian coordinates of the point.

Answer to Problem 3E

Hence, the required Cartesian coordinate point is $\sqrt{(-1,-\sqrt{3})}$

Explanation of Solution

Given:

Polar coordinates are : $\left(2,\frac{-2\pi }{3}\right)$

Calculation:

Consider the following point: $\left(2,\frac{-2\pi }{3}\right)$

Draw the following plot

  

Compare the point $\left(2,\frac{-2\pi }{3}\right)$ with the polar point $(r,o)$

Here, $r=2,\theta =\frac{-2\pi }{3}$

Recollect that

  $x=r\cos \theta ,y=r\sin \theta$

  $x=(2)\cos \left(\frac{-2\pi }{3}\right)$

  $=2\cos \left(\frac{2\pi }{3}\right)$

  $=2\left(-\cos \left(\frac{\pi }{3}\right)\right)$

  $=-2\left(\frac{1}{2}\right)$

  $=-1$

Continue the further simplification

  $y=(2)\sin \left(\frac{-2\pi }{3}\right)$

  $=-2\sin \left(\frac{2\pi }{3}\right)$

  $=-2\sin \left(\frac{\pi }{3}\right)$

  $=-2\left(\frac{\sqrt{3}}{2}\right)$

  $=-\sqrt{3}$

Hence, the required Cartesian coordinate point is $\sqrt{(-1,-\sqrt{3})}$

Conclusion:

Hence, the required Cartesian coordinate point is $\sqrt{(-1,-\sqrt{3})}$

(c)

To determine

To plot: the point and find the Cartesian coordinates of the point.

Answer to Problem 3E

Hence, the required Cartesian coordinate point is $\sqrt{(\sqrt{2},-\sqrt{2})}$

Explanation of Solution

Given:

Polar coordinates are: $\left(-2,\frac{3\pi }{4}\right)$

Calculation:

Consider the following point: $\left(-2,\frac{3\pi }{4}\right)$

Draw the following plot

  

Here, the point $\left(-2,\frac{3\pi }{4}\right)$ is located two units from the pole in the fourth quadrant because the angle $\frac{3\pi }{4}$ is in the second quadrant and $r=-2$ is negative.

Compare the point $\left(-2,\frac{3\pi }{4}\right)$ with the polar point $(r,\theta )$

Here, $r=-2,\theta =\frac{3\pi }{4}$

Recollect that

  $x=r\cos \theta ,y=r\sin \theta$

  $x=(-2)\cos \left(\frac{3\pi }{4}\right)$

  $=-2\left(-\cos \left(\frac{\pi }{4}\right)\right)$

  $=2\cos \left(\frac{\pi }{4}\right)$

  $=2\left(\frac{1}{\sqrt{2}}\right)$

  $=\sqrt{2}$

  $y=(-2)\sin \left(\frac{3\pi }{4}\right)$

  $=-2\sin \left(\frac{\pi }{4}\right)$

  $=-2\left(\frac{1}{\sqrt{2}}\right)$

  $=-\sqrt{2}$

Hence, the required Cartesian coordinate point is $\sqrt{(\sqrt{2},-\sqrt{2})}$

Conclusion:

Hence, the required Cartesian coordinate point is $\sqrt{(\sqrt{2},-\sqrt{2})}$