The parametric functions I am dealing with are: $x=2\sin2t$ and $y=3\sin t$
I know for a parametric graph to cross itself, there must be two distinct $t$, $t_1$ and $t_2$, that when placed into the two parametric functions, must produce the same ordered-pair; that is, $x=f(t_1)=f(t_2)$ and $y=g(t_1)=g(t_2)$.
With this, I have two system of equations:
(1) $2\sin 2t_1= 2\sin 2t_2 \implies 4\sin t_1\cos t_1=4\sin t_2\cos t_2 \implies \sin t_1=\dfrac{\sin t_2\cos t_2}{\cos t_1}$
(2) $3\sin t_1=3\sin t_2$
When I substituted $\sin t_1$ into (2), and simplified, I got $\cos t_2=\cos t_1$ Wouldn't that mean the graph intersects itself everywhere?