The polar coordinates of an ellipse are given by: $$x=\frac{abcos(\theta)}{\sqrt{b^2cos^2(\theta)+a^2sin^2(\theta)}}$$
$$y=\frac{absin(\theta)}{\sqrt{b^2cos^2(\theta)+a^2sin^2(\theta)}}$$
However, I have often seen the following being used in my textbook and other places: $$x=acos(\theta)$$ $$y=bsin(\theta)$$
Are these two forms the same? Is there something I'm unable to see? Or are there two ways to explain the coordinates of an ellipse?
The canonical equation for ellipse is:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
Check and see that both of these parametrizations satisfy the equation. So they both are correct in that sense.
But the angle dependense is not the same in two cases, meaning the first one is not the usual polar coordinate system.
For example, set $\theta=\frac{\pi}{4}$, you will get:
$$x=y=\frac{ab}{\sqrt{a^2+b^2}}$$
in the first case and:
$$x=\frac{a}{\sqrt{2}}$$
$$y=\frac{b}{\sqrt{2}}$$
in the second case.