The parametric curve given by x = cos at and y = sin bt is known as a Lissajous figure. The x-coordinate oscillates a times between 1 and -1 as t goes from 0 to 2ΰ, while the y-coordinate oscillates b times over the same t interval. This behavior is repeated over every interval of length 2π. The entire motion takes place in a unit square. Plot the following Lissajous figures for a range of t that ensures that the resulting figure is a closed curve. In each case, count the number of times that the curve touches the horizontal and vertical borders of the unit square.
(a) x = sin t, y = cos t
(b) x = sin 3t, y = cos 5t
(c) x = cos 5t, y = sin 15t
(d) x = sin 21, y = cos 9t