Converting $\frac{x^2}{4}+y^2\leq1$ to parametric form

Question

I have $\dfrac{x^2}{4}+y^2\leq1$ to be converted to parametric equation.

I have tried,

$x^2+4y^2\leq4$

$x^2\leq4(1-y^2)$

$x^2\leq4(1-y)(1+y)$

I am doubting for my next step since this is an inequality. Can you help me with this?

Answer 1

Hint:

Use the fact that $\cos^2(t) + \sin^2(t) = 1$ and set

$$ \begin{aligned} x^2/4 & = \cos^2(t) \\ y^2 & = \sin^2(t) \end{aligned}$$

Solve the above for $x(t)$ and $y(t)$ and prove that it solves the equation of the ellipse for all values of $t$.