Use
y=rsin(theta) x=rcos(theta)
Take dy/dx
Use the polar curve formula of dy/dx = rcos(theta) + sin(theta) (dr/d(theta)) / -rsin(theta)+cos(theta) (dr/d(theta))
So when your dx=0 you have a vertical tangent line.
Actually for this equations there are a lot of points where it should be vertical and the horizontal points aren't quite as easy because you have to solve something like
3(cos^2(theta)-sin^2(theta))=0