Chapter 11.3, Problem 4E

(a)

To determine

To find: The plot for each of the point with the given rectangular coordinates and find the two sets of corresponding polar coordinates.

Answer to Problem 4E

The required polar coordinates is $\left(-2,\frac{\pi }{6}\right),\left(2,\frac{7\pi }{6}\right)$ .

Explanation of Solution

Given Data:

The given coordinates is $\left(-\sqrt{3},-1\right)$ .

Calculation:

The point in the rectangular coordinate system is,

  $\begin{array}{c}r=\sqrt{{\left(-\sqrt{3}\right)}^2+{\left(-1\right)}^2}\\ =\pm 2\end{array}$

The phase angle is,

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{-1}{-\sqrt{3}}\right)\\ =\frac{\pi }{6},\frac{7\pi }{6}\end{array}$

Thus, the required polar coordinates is $\left(-2,\frac{\pi }{6}\right),\left(2,\frac{7\pi }{6}\right)$ .

(b)

To determine

To find: The plot for each of the point with the given rectangular coordinates and find the two sets of corresponding polar coordinates.

Answer to Problem 4E

The required polar coordinates is $\left(5,{\tan }^{-1}\left(\frac{4}{3}\right)\right),\left(-5,\pi +{\tan }^{-1}\left(\frac{4}{3}\right)\right)$ .

Explanation of Solution

Given Data:

The given coordinates is $\left(3,-4\right)$ .

Calculation:

The point in the rectangular coordinate system is,

  $\begin{array}{c}r=\sqrt{{\left(3\right)}^2+{\left(4\right)}^2}\\ =\pm 5\end{array}$

The phase angle is,

  $\theta ={\tan }^{-1}\left(\frac{4}{3}\right)$

Thus, the required polar coordinates is $\left(5,{\tan }^{-1}\left(\frac{4}{3}\right)\right),\left(-5,\pi +{\tan }^{-1}\left(\frac{4}{3}\right)\right)$ .

(c)

To determine

To find: The plot for each of the point with the given rectangular coordinates and find the two sets of corresponding polar coordinates.

Answer to Problem 4E

The required polar coordinates is $\left(2,0\right),\left(2,2\pi \right)$ .

Explanation of Solution

Given Data:

The given coordinates is $\left(2,0\right)$ .

Calculation:

The point in the rectangular coordinate system is,

  $\begin{array}{c}r=\sqrt{{\left(2\right)}^2+{\left(0\right)}^2}\\ =\pm 2\end{array}$

The phase angle is,

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{0}{2}\right)\\ =0,2\pi \end{array}$

Thus, the required polar coordinates is $\left(2,0\right),\left(2,2\pi \right)$ .

(d)

To determine

To find: The plot for each of the point with the given rectangular coordinates and find the two sets of corresponding polar coordinates.

Answer to Problem 4E

The required polar coordinates is $\left(-1,0\right),\left(1,\pi \right)$ .

Explanation of Solution

Given Data:

The given coordinate is $\left(-1,0\right)$ .

Calculation:

The point in the rectangular coordinate system is,

  $\begin{array}{c}r=\sqrt{{\left(-1\right)}^2+0^2}\\ =\pm 1\end{array}$

The phase angle is,

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{0}{1}\right)\\ =0,\pi \end{array}$

Thus, the required polar coordinates is $\left(-1,0\right),\left(1,\pi \right)$ .