At the moment I am working on the brachistochrone solved with the Euler Lagrange equation. The candidate one gets when using it is:
$$d^2=y(x)(1+(y'(x))^2)\text.$$
To find the solution (cycloid), I substituted with tangens, which is:
$${y'(x)=\tfrac{dy}{dx}=\tan(\alpha(x))=\sqrt{\tfrac{d^2-y(\alpha(x))}{y(\alpha(x))}}} \text{ and } {x \in [0,b]}\text.$$
Now I want to find the definition of range and definition of value for this substitution. My idea would be:
FIRST
$${x \mapsto \alpha(x)} \text{ with } {x \in [0,b]} \text{ and } {\alpha \in (-\tfrac{\pi}{2},\tfrac{\pi}{2})}\text.$$
THEN
$${\alpha \mapsto \tan(\alpha)} \text{ with } {\tan(\alpha) \in \mathbb{R}}\text.$$
Are these ranges correctly defined then? Thanks for all answers.