Chapter 9, Problem 45CLT

To determine

Calculate the three additional polar representation of the point and plot the given point.

Answer to Problem 45CLT

The three additional points are $(-2,-\frac{3\pi }{4}),(2,-\frac{\pi }{4})and(2,-\frac{7\pi }{4})$ .

Explanation of Solution

Given:

It is given in the question that the given point is $(-2,\frac{5\pi }{4})$ .

Concept Used:

In this, use the concept of using the formulas of polar representations: $(r,\theta )=(r,\theta +2n\pi )or(r,\theta )=(-r,\theta +(2n+1)\pi )$ .And plot the point in the graphing calculator.

Calculation: Here, the given point is $(-2,\frac{5\pi }{4})$ .

To find similar points within the interval use the formulas:

  $(r,\theta )=(r,\theta +2n\pi )or$

  $(r,\theta )=(-r,\theta +(2n+1)\pi )$

Using $(r,\theta )=(r,\theta +2n\pi )$ at $n=-1$ ,

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

Using $(r,\theta )=(-r,\theta +(2n+1)\pi )$ at $n=-1$ ,

  $\begin{array}{l}(-2,\frac{5\pi }{4})=(-2,\frac{5\pi }{4}-\pi )\\ (-2,\frac{5\pi }{4})=(2,-\frac{\pi }{4})\end{array}$

Using $(r,\theta )=(-r,\theta +(2n+1)\pi )$ at $n=-2$ ,

  $\begin{array}{l}(-2,\frac{5\pi }{4})=(-2,\frac{5\pi }{4}-3\pi )\\ (-2,\frac{5\pi }{4})=(2,-\frac{7\pi }{4})\end{array}$

Now, plot the given points on the polar representation in which $-2\pi <\theta <2\pi$

The graph is shown below:

  

Conclusion:

The three points are . $(-2,-\frac{3\pi }{4}),(2,-\frac{\pi }{4})and(2,-\frac{7\pi }{4})$