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?
 A: 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
$(\sin(t), \cos(t))$.
A: Because $e^{it} = \cos t+i\sin t$ (but not only because of that).
A: 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.
A: Assuming it is reasonable to view tan $\theta$ as more "natural" than cot $\theta$, then one advantage to making $x$ the cosine coordinate is it gives a direct correspondence between slopes and tangents.  But I much prefer the motivation in Michael Hardy's answer.
A: Using $t$ is the most ‘natural’  parametrisation  because $t$ is the polar angle  of the point $(x,y)$ in a standard polar coordinates system.
