Question

Let G be the graph of the parametric equations
x = cos(4t),
y = sin(6t).

What is the length of the smallest interval P such that the graph of these equations for all t E P produces the entire graph G?

Answer 1

if 4 t = 2 pi
t = pi/2
if 6 t = 2 pi
t = pi/3
so we need at least t = 0 to t = pi/2 or any other interval of length pi/2

Answer 2

In the interval -pi/4 <= t <= pi/4, the graph is traced once.

But on 0 <= t <= pi/2, only half the curve is completed.

Try changing the domain on this plot at

http://www.wolframalpha.com/input/?i=plot+x%3Dcos(4t),y%3Dsin(6t),+-pi%2F4+%3C%3D+t+%3C%3D+pi%2F4