Mirror, search and the inverse cotangent

You will write three small functions about inverse functions: the mirror image of a point, an inverse sine found by halving an interval, and the inverse cotangent. The checker tests them on many values.

The idea. A point $(a, b)$ of a function becomes $(b, a)$ on its inverse. On the sine branch the sine rises, so you can find the angle with a given sine by halving an interval again and again. The inverse cotangent with range $(0, \pi)$ is $\dfrac{\pi}{2} - \tan^{-1}x$.

Your tasks.

1. Finish mirror(point). It returns the point with its two numbers swapped.
2. Finish arcsin_by_search(v). Start with low = -pi / 2 and high = pi / 2. Halve the interval $60$ times, keeping the half where the sine of the angle can equal v.
3. Finish arccot(x). Return the principal value of the inverse cotangent, an angle in $(0, \pi)$.