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 of convension.
It is customary to view angles as lying with the vertex at the origin and the right leg along the positive $x$-axis. That means that small angles will have their left leg in the first quadrant, and right angles will have the left leg along the positive $y$-axis. Because of this, the cosine of the angle happens to be the $x$-coordinate of the point on the left leg that is distance $1$ from the vertex, while the sine of the angle is the $y$-coordinate.
