Parametric Equations

Consider the ellipse with rectangular equation

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

If $x=a\cos \theta$ and $y=b\sin \theta$ , then

$\frac{x^2}{a^2}+\frac{y^2}{b^2}= \frac{(a\cos \theta)^2}{a^2}+\frac{(b\sin \theta)^2}{b^2}=\cos^2\theta+\sin^2\theta=1$

and hence $(x,y)$ lies on the ellipse.

The parametric equations of the ellipse are given by $x = a\cos \theta$ , $y = b \sin\theta$ .