# Is $\|\sin{x}\|$ a cycloid?

Forgive this seemingly basic question; I recently found out about cycloids and cannot find any answers on the web. My guess is that it’s not, due to some part of the definition of a cycloid, but I can’t really find something about cycloids which explicitly helps me answer my question/I am not rounded enough to decipher the necessary information.

• It is not. ${}{}{}{}{}$ – Seewoo Lee Feb 23 at 23:29
• The derivative of the cycloid can take any real value, while the derivative of $| \sin x |$ is never outside the interval $[-1,1]$ when it exists. You will notice the difference in shape where $y=0$ – Henry Feb 23 at 23:39

One possible explanation is using the parametrization of a cycloid. It is given by $$x = r(t-\sin t)\\ y = r(1-\cos t)$$ for some $$r>0$$. If $$y = |\sin x|$$ is a cycloid, or more generally, homothetic to cycloid, then it will be possible to find some constant $$a\in \mathbb{R}$$ such that $$|\sin(r(t-\sin t))| = ar(1-\cos t)$$ for all $$t\in \mathbb{R}$$. However, it is not true.