[tex]3r^2-6r\sin\theta=0[/tex] is valid.
[tex]r^2-2r\sin\theta=0[/tex] is equally valid.
[tex]r-2\sin\theta=0[/tex] is not technically valid because it's possible that [tex]r=0[/tex], and you can't divide through the above equation by 0. The plots of all three of these options are identical, however.
