Chapter 8.4, Problem 52E

(a)

To determine

To find: the polar equation in parametric form.

Answer to Problem 52E

Therefore, the parametric form of the polar equation $r=2^{\sin \theta }$ is

  $\begin{array}{l}x=2^{\sin t}\cos t\\ y=2^{\sin t}\sin t\end{array}$

Explanation of Solution

Given:

polar equation $\begin{array}{l}r=2^{\sin \theta }\\ \end{array}$

Calculation:

Express the polar equation of the line $r=2^{\sin \theta }$ in the parametric form.

The graph of the polar equation $r=F\left(\theta \right)$ is the same as the graph of the parametric equation

  $x=F\left(t\right)\cos t$ ,and

  $y=F\left(t\right)\sin t$

Therefore, the parametric form of the polar equation $r=2^{\sin \theta }$ is

  $\begin{array}{l}x=2^{\sin t}\cos t\\ y=2^{\sin t}\sin t\end{array}$

Conclusion:

Therefore, the parametric form of the polar equation $r=2^{\sin \theta }$ is

  $\begin{array}{l}x=2^{\sin t}\cos t\\ y=2^{\sin t}\sin t\end{array}$

(b)

To determine

To find: the graph of the parametric equations.

Answer to Problem 52E

Therefore, the above equations in the graphing calculator,

  

Explanation of Solution

Given:

polar equation $r=2^{\sin \theta }$

Calculation:

Use a graphing device to graph the parametric equations $x=2^{\sin t}\cos t$

and $y=2^{\sin t}\sin t$ .

On entering the above equations in the graphing calculator ,

  

Conclusion:

Therefore, the above equations in the graphing calculator,