Chapter 10.3, Problem 32E

To determine

To find: The graph of the polar curve $r=-\csc \theta \cot \theta$ and identify the name of the curve.

Answer to Problem 32E

The graph of the curve is shown in Figure (1) and it is a parabola.

Explanation of Solution

Given information: The given equation is $r=-\csc \theta \cot \theta$ .

Calculation:

Simplify the equation.

  $\begin{array}{c}r=-\csc \theta \cot \theta \\ r=-\frac{1}{\sin \theta \tan \theta }\end{array}$

To graph a function $r=-\frac{1}{\sin \theta \tan \theta }$ , follow the steps using graphing calculator.

First press $\boxed{\text{MODE}}$ key on graphical calculator and select the polar mode. Press $\boxed{\text{Y}=}$ key and enter right hand side of the equation after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\left(-\right)}\boxed{1}\boxed{÷}\boxed{\text{SIN}}\boxed{\theta }\boxed{)}\boxed{\times }\boxed{\text{TAN}}\boxed{\theta }\boxed{)}$ .

The display will show the equation,

  ${\text{r}}_{\text{1}}=-1/\left(\sin \left(\theta \right)*\tan \left(\theta \right)\right)$

The appropriate viewing window for the function is $\left[-10,10\right]$ by $\left[-10,10\right]$ .

Now, press the $\boxed{\text{GRAPH}}$ key to produce the graph of given equation as shown in Figure (1).

  

Figure (1)

Thus, the graph of the curve is shown in Figure (1) and it is a parabola.