Deven Livingston

2022-05-02

Showing surjectivity of $f\left(x\right)=\frac{x-\mathrm{sin}x}{1-\mathrm{cos}x}$

is strictly monotonically increasing on$x\in (0,2\pi )$ , I'd like to show the surjectivity of this function [to $(0,\mathrm{\infty})$ ]. However, I fail solving the equation

$y=\frac{x-\mathrm{sin}\left(x\right)}{1-\mathrm{cos}\left(x\right)}$

with respect to x. Can I somehow use/apply the fact that$f\left(x\right)=\frac{u\left(x\right)}{{u}^{\prime}\left(x\right)}$ ?

is strictly monotonically increasing on

with respect to x. Can I somehow use/apply the fact that

Leia Wiggins

Beginner2022-05-03Added 18 answers

Just show that the function is continuous in the domain, and since it tends to 0 as $x\to 0$ , and tends to $\mathrm{\infty}$ as $x\to 2\pi$ , you are done as you have already shown it's monotonically increasing in the domain.

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