Different parametric equation for a circle

I found this one:

an animation:

I found it by using the first couple of terms for the power series of sine and cosine, and making that the function for my x and y values, dividing each by the magnitude of x and y together. Then rotating the function and simplifying.

So all the points are exactly on the unit circle, but I noticed it doesn't go around always at the same speed, for example t=.5 or -.5 gives x value ~.2626 instead of 0. I'm looking at whether one could substitute in a function s for t that would make for even speed around the circle...