Equation tangent line question?

Question

How do you know where the curves will cross? For example: The problem "x=2 sin 2t, y=3 sin t" crosses itself at the origin, (0,0). "At this point, t=0 or t=pi." How do you know this?

Answer 1

solve the equations
x=2sin2t----(1)
y=3sint------(2)
to get values of x & y.

These (x,y) are the points where these curves will meet/cross.

from (1), since y component is not there means y=0,
from (2), since x component is not there means x=0.

Therefor, these curves meet at (0,0).
Now from (2), since y=0 at (0,0),
3sint=0
sint=0
t=0 or pi

Answer 2

2sin2t= 3sint
as sin 2t= 2 sint cos t
sin t ( 4cost -3)=0 so sin t =0 and t = kpi ( k= any integer)
and
cos t =3/4 so t=0.7227+2kpi and t =5,5605+2kpi