A parametric curve is a pair of functions x=f(t), y=g(t) that give x and y as contiuous functions of the real number t (the parameter). Each value determines a point ( f(t), g(t) ). The graph is the set of all such points in the interval assigned.
The point (cos t, sin t) begins at (1, 0). As t moves counterclockwise from 0 to 2 pi, it gives points that satisfy: x = cos t, y = sin t
All points lie on the unit circle, thus the graph is the unit circle.
This parametric curve also lies on the unit circle because
To explore other graphs I changed the parameters:
For various a and b, x = cos (at) ; y = sin (bt)
Changes the period.
For various a and b, x = a cos (t) ; y = b sin (t)
Changes the amplitude. In general, when a and b increase, the amplitude increases. When a and b decreases, the period decreases.
a is the amplitude and 2pi/b is the period.