TITLE: Showing surjectivity of $f(x) = \frac{x - \sin x}{1 - \cos x}$
QUESTION [6 upvotes]: After having shown that 
$$f(x) = \frac{x - \sin (x)}{1 - \cos (x)}$$
is strictly monotonically increasing on $x \in (0, 2\pi)$, I'd like to show the surjectivity of this function [to $(0,\infty)$]. However, I fail solving the equation 
$$y =\frac{x - \sin (x)}{1 - \cos (x)}$$
with respect to $x$. Can I somehow use/apply the fact that $f(x) = \frac{u(x)}{u'(x)}$ ? I recognized this, even though I could not use it proving the monotonicity, so maybe I can use it here. For any hints I am very thankful.

REPLY [9 votes]: Just show that the function is continuous in the domain, and since it  tends to $0$ as $x \rightarrow 0$, and tends to $\infty$ as $x \rightarrow 2\pi$, you are done as  you have already shown it's monotonically increasing in the domain.