show us your effort/s to solve this problem.
From simple trigonometry I know that \(\displaystyle \cos^2 \theta + \sin^2 \theta = \sin^2 \theta + \cos^2 \theta = 1.\) So the best parametrization to meet the given condition is to choose:
\(\displaystyle x(t) = \sin t\)
\(\displaystyle y(t) = \cos t\)
Then, our parametrized curve is:
\(\displaystyle \alpha(t) = (x(t), y(t)) = (\sin t, \cos t)\)
Let us test the given condition.
\(\displaystyle \alpha(0) = (\sin 0, \cos 0) = (0,1)\), it is indeed satisfied.
Let us also check if the trace runs clockwise. If at \(\displaystyle t = 0\), we have the point \(\displaystyle (0,1)\) (at the top of the circle), I expect at \(\displaystyle t = 1\), the \(\displaystyle x\) increases and the \(\displaystyle y\) decreases. This will make a particle, for example, at the top of the circle to move clockwise into quadrant \(\displaystyle \text{I}\).
\(\displaystyle \alpha(1) = (\sin 1, \cos 1) = (0.84,0.54)\)
This point is indeed in the first quadrant
then the curve runs clockwise.
