I'm looking for a function whos graph looks like a cycloid of a circle with the radius 1. Given the parametric equations for this cycloid: $$\begin{align}x& = t - \sin(t)\\y &= 1 - \cos(t)\end{align}$$
From this, one can find an equation for $t$:$$t = x + \sin(t)$$
This equation for $t$ can be substituted for $t$ in the equation for $y$:$$y = 1-\cos(x+\sin(t))$$
Yet $t$ still remains in the equation for $y$. To get an equation for $y$ in terms of $x$ one can substitute $t$ infinitely many times:
$$y=1-\cos(x+\sin(x+\sin(x+\sin(x+\sin(x+\sin(\cdots))))))$$
Is there a way to resolve these infinetely nested sin functions?