Converting $x=\frac{1}{2}\cos\theta\;;\;\; y=2\sin\theta $ to Cartesian form

Question

How can we transform these parametric equations to Cartesian form?

$x=\frac{1}{2} \cos\theta, \quad y=2\sin\theta \quad\text{ for}\;\;0 \leq \theta \leq \pi$

Answer 1

$(2x)^2+(y/2)^2=\sin^2(\theta)+\cos^2(\theta)=1 $

Answer 2

We can use the fact that $\sin^2(\theta) + \cos^2(\theta) = 1,$ and then express the given parametric equations in terms of $\cos\theta$ and $\sin\theta$, respectively.

$x=\frac{1}{2}\cos\theta \iff 2x = \cos\theta$

$y=2\sin\theta \iff \frac12y=\sin\theta$.

Now using the identity: $$\sin^2(\theta) + \cos^2(\theta) = 1,$$ you simply need to substitute into the identity the expressions above for $\sin\theta$ and $\cos\theta$.

Doing so will give you a Cartesian function in $x$ and $y$.