Let's say I want to know the vector function that represents the intersection of the two surfaces for a cylinder $x^2+y^2=4$ and $z=xy$
Would it be alright for me to declare that $x=\sin(t)$ and not $x=\cos(t)$? I had this question because my textbook always assumes the latter and not the former. Furthermore, I was also wondering if it would be alright for me to arbitrarily declare $x=t$ and continue deducing what the other variables' parameterization could be?
*edit
I meant to say $2\sin(t)$ and $2\cos(t)=x$
You can choose whichever parameterization you'd like, but it's important to then keep in mind the direction.
Notice that under the `standard' parameterization $x = 2\cos(t), y =2\sin(t)$ you travel around the circle counter-clockwise (between $t = 0, \pi/2$ you travel from $(2,0)$ to $(0,2)$); however, under your parameterization $x = 2\sin(t), y = 2\cos(t)$ when $t = 0$ you start at $(0,2)$ and the proceed clockwise around the circle.
This is not inherently a problem, but occasionally things can differ slightly.
e.g. The helix $x = \sin(t), y = \cos(t), z = t$ is actually distinct from $x = \cos(t), y=\sin(t), z =t$.
(Look at the intersection with the $xy$-plane.)