When graphing, we are comparing two variables (usually x and y). Parametric equations connect these two variables through a third variable, the parameter (often designated p, t or theta). To graph parametric equations, we need to combine them in such a way as to eliminate the parameter and produce a single Cartesian equation. Here's a classic example:
x=cost -> (1)
y=sint -> (2)
Square both equations:
x^2=cos^2t -> (1a)
y^2=sin^2t -> (2a)
Now add equations (1a) and (2a):
x^2+y^2=cos^2t+sin^2t
x^2+y^2=1 -> (3)
Now we've eliminated the parameter, t. So, equation (3) is the Cartesian equation that corresponds to the original parametric equations, (1) and (2). Incidentally, this shows that using the basic trigonometric functions as the basis for our parametric equations produces the unit circle.
