Question

1. Investigate the family of curves defined by the polar equations \(r = \sin n\theta \) , where \(n\) is a positive integer. How is the number of loops related to \(n\) ?
2. What happens if the equation in part(a) is replaced by\(r = \left| {\sin n\theta } \right|\)?

Short answer

Number of loops is always \(2n\) .

Step-by-step solution

Step 1:

(a)

Consider the polar equation \(r = \sin n\theta \) , where \(n\) is a positive integer.

The number of loops are \(n\) , when \(n\) is odd and the number of loops are \(2n\) , when \(n\) is even.

For example \(r = \sin \left( {4\theta } \right)\)

Consider the graph of \(r = \sin \left( {4\theta } \right)\) .

And for an odd multiple of \(n\) we have

Step 2:

(b)

Now when we have the following

\(r = \left| {\sin \left( {n\theta } \right)} \right|\)

This will have \(2n\) loops whether \(n\) is odd or even since we have the period every \(\pi \) .

Number of loops is always \(2n\) .