Find $k$ such that $-1/2<\cos(-(2π/n)×k+(2π/n)×(n+1)/2)<1/2$

Question

Let $n>6$ be an odd positive integer. I come across with this inequality:

Can we find a positive integer $k$ in the set $1,2,...,(n-1)/2$ such that $$-1/2<\cos(-(2π/n)×k+(2π/n)×(n+1)/2)<1/2$$

Answer 1

So you are looking for $k\in\{1,2,...{n-1\over 2}\}$ such that (since the function $arccos$ is decreasing on the interval $(π/3, 2π/3)$ $${\pi\over 3}<-{2π\over n}×k+{2π\over n}×{n+1\over 2}<{2\pi\over 3}$$ which is the same as (if we cancel $\pi$ and multiply by $n$)

$${n\over 3}<-{2k}+{n+1}<{2n\over 3}$$ or $$n<-{6k}+{3n+3}<{2n}$$

or $$\color{red}{0}<{n+3\over 6}< k<{2n+3\over 6}<\color{red}{n-1\over 2}$$ if $n>6$

So the answer is: yes.