To determine the trajectory of the movement of a material point in the xoz plane that is subjected simultaneously to oscillations x (t) = a * sin π(t + 1) and y (t) = a * sin (πt + π / 2).

We are to show that the given parametric curve is a circle.

The trajectory of a circle with a radius r will satisfy the following relationship:

(with (x_c,y_c) being the center point)

We are given the x and y in a parametric form which can be further rewritten (using properties of sin/cos):

Squaring and adding both gives:

The last expression shows that the given parametric curve is a circle with the center (0,0) and radius A.