Rotation equivalent to what two reflections

Question

What two reflections are equivalent with the rotation about $(0,-1)$ by $180^\circ$?

Can I say one of the reflections is about the $y$-axis? If so, what would the second reflection be? Would it be $y=-1$?

Answer 1

Consider a point given by $re^{i\theta}$ to be rotated around the origin (first translate the point by rotational origin minus actual origin, then rotate, and then translate the new point back).

Rotation is given by $re^{i(\pi+\theta})$.

Reflection in the $x$-axis is $re^{-i\theta}$.

Reflection in the $y$-axis is $re^{i(\pi-\theta)}$.

Putting the two reflections together gives $re^{i(-\pi+\theta)}$

which is the same as the rotation because $e^{i\pi}=e^{-i\pi}=-1$.

So your two reflections are about the $x$-axis and $y$-axis, translated to the point $(0,-1)$, i.e. the lines of reflection are $x=0, y=-1$.