(a) Consider the family of curves given by the polar equations $r=\sin(n\theta),$ where $n$ is a positive integer. How is the number of loops related to $n$? Check all that apply.

A. There are $2n$ loops when $n$ is even.
B. There are $4n$ loops when $n$ is even.
C. There are $4n$ loops when $n$ is odd.
D. There is exactly 1 loop for each $n$.
E. There are no loops.
F. There are $n$ loops when $n$ is odd.
G. There are $n$ loops when $n$ is even.
H. There are $2n$ loops when $n$ is odd.

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(b) Now consider the family of curves given by the polar equations $r=|\sin(n\theta)|,$ where $n$ is a positive integer. How is the number of loops related to $n$? Check all that apply.

A. There are $4n$ loops when $n$ is even.
B. There is exactly 1 loop for each $n$.
C. There are $4n$ loops when $n$ is odd.
D. There are $n$ loops when $n$ is even.
E. There are $2n$ loops when $n$ is odd.
F. There are $2n$ loops when $n$ is even.
G. There are $n$ loops when $n$ is odd.
H. There are no loops.

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