For an ellipse defined by a general quadratic equation in $x$ and $y$, the trigonometric parametrization is obtained. The conic determined by this equation is graphed, and so is the parametrized version.

The graphs serve as validation of the parametrization - if the graph of the original quadratic and the graph of the parametric form coincide, then that is taken as evidence that the parametrization is correct.

Student[Precalculus][ConicsTutor], geometry[ellipse], geometry[detail]