Inverse of a curve

Question

I'm trying to invert a function from $\mathbb{R}\rightarrow\mathbb{R}^3$ and don't know where to begin, for example suppose this one $$\alpha(t)=(\cos t,\operatorname{sen} t,t),$$ does it have inverse? How can i get to $\mathbb{R}^3\rightarrow\mathbb{R}$ or $\alpha(\mathbb{R})\rightarrow\mathbb{R}$? I thought of things like the one below, but I don't know how to proceed. $$ \begin{array} tt=\cos^{-1}x\\ t=\operatorname{sen}^{-1}y\\ t=z \end{array} $$ How can I unify these equations or even check if the inverse really works?