Chapter 11.3, Problem 18E

To determine

To determine: The appropriate window to produce the polar curve. Also, sketch the complete curve and identify the type of the curve by name.

Answer to Problem 18E

The curve is a Lemniscate.

Explanation of Solution

Given information:

The equation in polar coordinate system: $r^2=\sin 2\theta$

Calculation:

The given equation is $r^2=\sin 2\theta$ .

Rewrite the above expression as:

  $\begin{array}{c}\sqrt{r^2}=\sqrt{\sin 2\theta }\\ r=\pm \sqrt{\sin 2\theta }\end{array}$

From the above equation, it is seen that $\left|r\right|=\left|\pm \sqrt{\sin 2\theta }\right|\le 1$ .

So, consider the point $\left[-1,1\right]$ for y -range and $\left[-1.5,1.5\right]$ for the x -range.

The graph of the given equation is shown below:

  

From the above graph it is seen that the curve is a Lemniscate.