Question

Graphing Polar Equations

Use a graphing device to graph the polar equation. Choose the domain $\theta$ to make sure you produce the entire graph

$r=\sin \left(\frac{8\theta }{5}\right)$

Short answer

The graph of the given polar equation is-

Step-by-step solution

Step 1. Given information

A polar equation is given as $r=\sin \left(\frac{8\theta }{5}\right)$

Step 2. Concept used

For sketching the graph, first, find different values r for different$\theta$. After that, convert them into x and y by using equations-

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

Once we have the points draw a graph by plotting them.

Step 3. Calculation

The polar equation is given as$r=\sin \left(\frac{8\theta }{5}\right)$,

Let$\theta =0$,

$\begin{array}{l}r=\sin \frac{8\theta }{5}\\ r=\sin \left(0\right)\\ r=0\end{array}$

Now when we plug the values in the equations $\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

We get, $\begin{array}{l}x=0\\ y=0\end{array}$

Similarly,

Let$\theta =1$,

$\begin{array}{l}r=\sin \frac{8\theta }{5}\\ r=0.99\end{array}$

Plot the points on graph-