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.
