Chapter 12, Problem 6CLT

To determine

Find three additional polar representations of the points.

Answer to Problem 6CLT

The three equivalent points are $\left(2,\frac{\pi }{4}\right)$ , $\left(2,\frac{-7\pi }{4}\right)$ , $\left(-2,\frac{5\pi }{4}\right)$

Explanation of Solution

Given information:

Plot the point $\left(-2,-3\pi /4\right)$ and find three additional polar representations of the point, using $-2\pi <\theta <2\pi$ .

Calculation:

Consider the given polar coordinates,

  $\left(-2,-3\pi /4\right)$

Find the equivalent points.

Replace $r$ by $-r$ and add $\pi$ to $\theta$ .

  $\begin{array}{l}\left(-2,-3\pi /4\right)=\left(2,\pi +-3\pi /4\right)\\ =\left(2,\pi /4\right)\end{array}$

Now plot the point on polar plane by using Use graphic utility $TI-83$ calculator to draw the graph.

Press mode and select Pol. The display will be,

  

Press $Y=$ and type the value of $r$ . The display will be,

  

Now press window and select proper scale,

  

Now press graph.

  

Now locate the point by moving an angle $\pi /4$ from $x-axis$ .

  

The equivalent points are,

Replace $r$ by $-r$ and add $\pi$ to $\theta$ .

  $\begin{array}{l}\left(-2,-3\pi /4\right)=\left(2,\pi +-3\pi /4\right)\\ =\left(2,\pi /4\right)\end{array}$

  $\left(2,\frac{\pi }{4}\right)$

Second equivalent points are,

Replace $r$ by $-r$ and subtract $\pi$ to $\theta$ .

  $\begin{array}{l}\left(-2,-3\pi /4\right)=\left(2,-\pi +-3\pi /4\right)\\ =\left(2,-7\pi /4\right)\end{array}$

  $\left(2,\frac{-7\pi }{4}\right)$

Third equivalent points are, add $2\pi$ to $\theta$ .

  $\begin{array}{l}\left(-2,-3\pi /4\right)=\left(2,2\pi +\frac{-3\pi }{4}\right)\\ =\left(-2,\frac{5\pi }{4}\right)\end{array}$

  $\left(-2,\frac{5\pi }{4}\right)$

Hence the three equivalent points are $\left(2,\frac{\pi }{4}\right)$ , $\left(2,\frac{-7\pi }{4}\right)$ , $\left(-2,\frac{5\pi }{4}\right)$