The point where r(t) intersects the xy-plane means the z component is zero. So let's set it to zero and we have:
sin(10t)cos(20t) = 0
To get zero value, either one of them can be zero:
sin(10t) = 0 or cos(20t) = 0
Sine function is zero when the following condition is satisfied:
10t = n*pi, where n = 0,1,2,...
Meanwhile cosine function is zero when:
20t = m*pi/2, where m = 1,3,5,7,...
The rest should be straightforward.