How to derive the cartesian equation of cycloid expressed for y?

Question

I found an expression for x in wiki: https://proofwiki.org/wiki/Equation_of_Cycloid_in_Cartesian_Coordinates but I need it expressed for y. How can I do that?

Answer 1

It cannot be done with only elementary functions. The closest you can do is define function $$ F(\theta) = \theta - \sin\theta $$ and use its inverse $F^{-1}$ (but that inverse cannot be expressed using only elementary functions and algebraic operations, see Inverse of $f(x)=\sin(x)+x$ ). We have then $$ y(x) = a\Big(1-\cos \big( F^{-1}(\frac{x}{a}) \big)\Big) $$ It may be still useful, depending on what do you want to obtain from it, because although $F^{-1}$ cannot be expressed using elementary functions, it can be still examined using the tools of mathematical analysis.