Chapter 6.4, Problem 71E

Solution

To write: The parametric equations for the given polar equation.

The parametric equations are $x=4\cot \theta$ and $y=4$ .

Given information:

The polar equation is $r=4\csc \theta$ .

Formula used:

The rectangular coordinate of the polar coordinate $\left(r,\theta \right)$ is given by,

  $\begin{array}{c}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Calculation:

The parametric equations for $r=f\left(\theta \right)$ are $x=f\left(\theta \right)\cos \theta$ and $y=f\left(\theta \right)\sin \theta$ .

From the given polar equation,

  $f\left(\theta \right)=4\csc \theta$

Substitute $f\left(\theta \right)=4\csc \theta$ in equation $x=f\left(\theta \right)\cos \theta$ .

  $\begin{array}{c}x=4\csc \theta \cos \theta \\ =\frac{4}{\sin \theta }\cos \theta \\ =4\cot \theta \end{array}$

Substitute $f\left(\theta \right)=4\csc \theta$ in equation $y=f\left(\theta \right)\sin \theta$ .

  $\begin{array}{c}y=4\csc \theta \sin \theta \\ =\frac{4\sin \theta }{\sin \theta }\\ =4\end{array}$

Draw the graph of polar equations in the viewing rectangle of $\left[-10,10\right]$ by $\left[-10,10\right]$ .

  

Draw the graph of parametric equations in the viewing rectangle of $\left[-10,10\right]$ by $\left[-10,10\right]$ .

  

From the above graph, it can be observed that both the graphs are equivalent.

Therefore, the parametric equations are $x=4\cot \theta$ and $y=4$ .