A curve has parametric equations x = 1 - cos(t), y = sin(t)sin(2t) for 0 <= t <= pi. Find the coordinates where the curve meets the x-axis.

If the curve is meeting the x-axis, notice that this means y = 0. So we must solve sin(t)sin(2t) = 0 for t within the given bounds. Using a trigonometric identity sin(2t) = 2cos(t)sin(t), we obtain sin²(t)cos(t) = 0. That is, EITHER sin²(t) = 0 meaning sin(t) = 0, or cos(t) = 0. We have to be slightly careful to keep t within the bounds. If sin(t) = 0 then t = 0 or t = pi. If cos(t) = 0 then t = pi/2, giving us 3 solutions in total. From here we simply substitute in our values of t. So x = 1 - cos(0) = 0 and y = sin(0)sin(0) = 0 giving us the points (0,0) when t = 0. Secondly, x = 1 - cos(pi/2) = 1 and y = sin(pi/2)sin(pi) = 0 since sin(pi) = 0 and so we get (1,0) when t = pi/2. Finally, x = 1 - cos(pi) = 2 and y = sin(pi)sin(2pi) = 0 and so we get (2,0) when t = pi.