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:
[tex](x-x_c)^2 + (y-y_c)^2 = r^2[/tex]
(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):
[tex]x(t) = A\sin \pi(t+1) = A\sin (\pi t + \pi) = -A\sin \pi t\\y(t) = A\sin (\pi t + \frac{\pi}{2}) = A\cos \pi t[/tex]
Squaring and adding both gives:
[tex]x^2(t) + y^2(t) = A^2(-1)^2\sin^2 \pi t + A^2 \cos^2 \pi t = A^2\\\implies x^2 + y^2 = A^2[/tex]
The last expression shows that the given parametric curve is a circle with the center (0,0) and radius A.
