So I have some parametric functions (of one variable) I'm trying to sketch.
Generally I can do so by "reverse parametrizing" where I take $x(t)$ and make $t$ a function of $x$ and then substituting that into $y(t)$ and end up with an original valid function $y(x)$. The problem is that this method isn't very general and there are parametric equations that are not valid functions or the math becomes far too tricky.
Another method is to make a table of $t$, $x$ and $y$, plug in the appropriate values of $t$ given some domain (e.g. $0<t<1$) and solve for the other two. The problem with this method is that it can be hard when you have domains of sufficiently small size or when stuff like square roots and logarithms become involved forcing me to work with non-integers and needing a calculator for small values of $t$.
Here is one such problem:
SKETCH: $$x(t)=2sin(t)$$ $$y(t)=cos(t)$$ for $0\le t \le 1$
From the equations, this looks like an ellipse so how can I "complete the square" for such parametrizations and get it into this familiar form:
$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$$
(it is an ellipse right?)
Hint: from your equations $x(t)=...$ and $y(t)=...$, try to solve for $\sin t$ and $\cos t$. Then try to use $\sin^2 t+\cos^2 t=1$ to get rid of parameter $t$.
EDIT: From $x(t)=2\sin t$ and $y(t)=\cos t$, we obtain $\sin t=x(t)/2$ and $\cos t=y(t)$. Thus $$1=\sin^2 t+\cos^2 t=(x(t)/2)^2+y(t)^2=1 \implies (x/2)^2+y^2=1$$