Chapter 8, Problem 26RE

To determine

To graph:

The equation of $r=\sin \left(\frac{9\theta }{4}\right)$ is need to drawn.

Explanation of Solution

Given:

  $r=\sin \left(\frac{9\theta }{4}\right)$

Graph:

  

Interpretation:

From the given data:

The given equation is in the polar co-ordinates is drawn as above:

  $r=\sin \left(\frac{9\theta }{4}\right)$

In order to view the full graph of the given equation, need to estimate the value of $\left(\theta \right)$ at which given function $\sin \left(\frac{9\theta }{4}\right)$ begins to repeat.

This value is nothing else than the period of $\sin \left(\frac{9\theta }{4}\right)$ which can be estimated by dividing $2\pi$ by the coefficient of the value inputted to sin function $\left(\frac{9}{4}\right)$

Period is tabulated as:

  $\begin{array}{l}P=\frac{2\pi }{\left(\frac{9}{4}\right)}\\ P=\frac{8\pi }{9}\end{array}$

So, the period value is $\frac{8\pi }{9}$ . Therefore, the value of $\theta$ must lie between $0$ and $6\pi$ as $0\le \theta \le \frac{8\pi }{9}$ .

So, through angle range $0\le \theta \le \frac{8\pi }{9}$ , the given equation is fully graphed which is shown by the above graph.

  $r=\sin \left(\frac{9\theta }{4}\right)\left\{0\le \theta \le \frac{8\pi }{9}\right\}$