Chapter 6.5, Problem 6E

Solution

To find: The smallest $\theta$ interval that will complete the graph and to plot the rose curve.

The smallest $\theta$ interval that will complete the graph is $k=\frac{\pi }{5}$ .

Given information:

The given expression is $r=3\sin 5\theta$ .

Calculation:

The polar equation of a rose curve is $r=a\cos n\theta$ or $r=a\sin n\theta$ . The rose curve has $n$ petals if $n$ is odd or has $2n$ petals if $n$ is even.

Here $n=5$ so the graph to have $5$ petals.

The plot of the rose curve represented by $r=3\sin 5\theta$ is shown below.

  

Figure (1)

To complete the graph start from $\theta =0$ in a graphing calculator. End the interval with a rational multiple of $\pi$ . Start with $k=\frac{\pi }{5}$ and increased its value until a complete graph is obtained.

In this process, one complete graph is produced for $k=\frac{\pi }{5}$ .