Why do we take $x=\cos t$, $y=\sin t$ for a parametric circle when we can take the opposite? Both $x = \cos t$, $y = \sin t$ and $x = \sin t$, $y = \cos t$ describe a circle

So why is the first parameterization so commonly used in mathematics, and not the second?
Because we usually
start the circle
at the right.
There,
the coordinates are
$(1, 0)$,
which is
$(\cos(0), \sin(0))$.

If we started it at the top,
we would probably use
$(\sin(t), \cos(t))$.
Using $t$ is the most ‘natural’  parametrisation  because $t$ is the polar angle  of the point $(x,y)$ in a standard polar coordinates system.