Chapter 11.3, Problem 2E

(a)

To determine

To determine: The rectangular coordinates of the given polar coordinates. Also, plot each point with the given polar coordinates.

Answer to Problem 2E

The required corresponding point in the rectangular coordinate system is $\left(\frac{3\sqrt{3}}{2},\frac{-3}{2}\right)$ .

Explanation of Solution

Given information:

The polar coordinate system: $\left(-3,\frac{5\pi }{6}\right)$

Calculation:

The given polar coordinate system is $\left(-3,\frac{5\pi }{6}\right)$ .

It is known that in a polar coordinate system if $r=f\left(\theta \right)$ is a curve, then the coordinates of the corresponding rectangular coordinate system are,

  $x=r\cos \theta =f\left(\theta \right)\cos \theta$ and $y=r\sin \theta =f\left(\theta \right)\sin \theta$

Write the above coordinates as:

  $\begin{array}{c}x=\left(-3\right)\cos \left(\frac{5\pi }{6}\right)\\ =\frac{3\sqrt{3}}{2}\end{array}$

And, $\begin{array}{c}y=\left(-3\right)\sin \left(\frac{5\pi }{6}\right)\\ =\frac{-3}{2}\end{array}$

Thus, the required corresponding point in the rectangular coordinate system is $\left(\frac{3\sqrt{3}}{2},\frac{-3}{2}\right)$ .

The plotting of the point is shown below:

  

Hence, the required corresponding point in the rectangular coordinate system is $\left(\frac{3\sqrt{3}}{2},\frac{-3}{2}\right)$ .

(b)

To determine

To determine: The rectangular coordinates of the given polar coordinates. Also, plot each point with the given polar coordinates.

Answer to Problem 2E

The required corresponding point in the rectangular coordinate system is $\left(3,4\right)$ .

Explanation of Solution

Given information:

The polar coordinate system: $\left(5,{\tan }^{-1}\left(\frac{4}{3}\right)\right)$

Calculation:

The given polar coordinate system is $\left(5,{\tan }^{-1}\left(\frac{4}{3}\right)\right)$ .

It is known that in a polar coordinate system if $r=f\left(\theta \right)$ is a curve, then the coordinates of the corresponding rectangular coordinate system are,

  $x=r\cos \theta =f\left(\theta \right)\cos \theta$ and $y=r\sin \theta =f\left(\theta \right)\sin \theta$

Write the above coordinates as:

  $\begin{array}{c}x=5\cos \left({\tan }^{-1}\left(\frac{4}{3}\right)\right)\\ =5\left(\frac{3}{5}\right)\\ =3\end{array}$

And, $\begin{array}{c}x=5\sin \left({\tan }^{-1}\left(\frac{4}{3}\right)\right)\\ =5\left(\frac{4}{5}\right)\\ =4\end{array}$

Thus, the required corresponding point in the rectangular coordinate system is $\left(3,4\right)$ .

The plotting of the point is shown below:

  

Hence, the required corresponding point in the rectangular coordinate system is $\left(3,4\right)$ .

(c)

To determine

To determine: The rectangular coordinates of the given polar coordinates. Also, plot each point with the given polar coordinates.

Answer to Problem 2E

The required corresponding point in the rectangular coordinate system is $\left(1,0\right)$ .

Explanation of Solution

Given information:

The polar coordinate system: $\left(-1,7\pi \right)$

Calculation:

The given polar coordinate system is $\left(-1,7\pi \right)$ .

It is known that in a polar coordinate system if $r=f\left(\theta \right)$ is a curve, then the coordinates of the corresponding rectangular coordinate system are,

  $x=r\cos \theta =f\left(\theta \right)\cos \theta$ and $y=r\sin \theta =f\left(\theta \right)\sin \theta$

Write the above coordinates as:

  $\begin{array}{c}x=\left(-1\right)\cos \left(7\pi \right)\\ =1\end{array}$

And, $\begin{array}{c}x=\left(-1\right)\sin \left(7\pi \right)\\ =0\end{array}$

Thus, the required corresponding point in the rectangular coordinate system is $\left(1,0\right)$ .

The plotting of the point is shown below:

  

Hence, the required corresponding point in the rectangular coordinate system is $\left(1,0\right)$ .

(d)

To determine

To determine: The rectangular coordinates of the given polar coordinates. Also, plot each point with the given polar coordinates.

Answer to Problem 2E

The required corresponding point in the rectangular coordinate system is $\left(-\sqrt{3},3\right)$ .

Explanation of Solution

Given information:

The polar coordinate system: $\left(2\sqrt{3},\frac{2\pi }{3}\right)$

Calculation:

The given polar coordinate system is $\left(2\sqrt{3},\frac{2\pi }{3}\right)$ .

It is known that in a polar coordinate system if $r=f\left(\theta \right)$ is a curve, then the coordinates of the corresponding rectangular coordinate system are,

  $x=r\cos \theta =f\left(\theta \right)\cos \theta$ and $y=r\sin \theta =f\left(\theta \right)\sin \theta$

Write the above coordinates as:

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

And, $\begin{array}{c}y=\left(2\sqrt{3}\right)\sin \left(\frac{2\pi }{3}\right)\\ =3\end{array}$

Thus, the required corresponding point in the rectangular coordinate system is $\left(-\sqrt{3},3\right)$ .

The plotting of the point is shown below:

  

Hence, the required corresponding point in the rectangular coordinate system is $\left(-\sqrt{3},3\right)$ .