I have the equation $4x^2+y^2=4$.
My instinct when parametrizing it was to do $2\cos(t)=x$ and $\sin(t)=y$ because the 4 is attached to the x. But, as you probably know it is really $x=\cos(t)$ and $y=2\sin(t)$, which is very clear from a graph.
My question is - why is this? Meaning - what is the intuition behind this fact?
Rewrite $4x^2+y^2=4$ as
$$x^2+(\frac y2)^2=1$$ Compare with $\cos^2t+\sin^2t=1$ to have the parametrization
$$x=\cos(t),\>\>\>\>\>\frac y2=\sin(t)$$