I'm trying to determine the order of growth of the function
\begin{align*} k \mapsto \min \lbrace s > 0 : 2\cdot \cos(\tfrac{s}{2})^{2(k-2)} - \sin(\tfrac{s}{2}) - 1 = 0 \rbrace, \quad k \geqslant 3 \text{ an integer.} \end{align*} However, I must say that I'm not really sure how to even begin... Any thoughts on this?
Motivation: In relation to a project I'm working on, I'm trying to understand the images $\Omega_k$ of a certain family of maps
\begin{align*} J_k : (0, \pi)^{{k}\choose{2}} \rightarrow (0, \pi)^{{k}\choose{2}}, \end{align*} mapping a bunch of angles to another bunch of angles. (I'll spare you the details as they're not much to look at...) The question above turns out to be related to the existence of some nice sets (i.e. cubes) inside the quite strange-looking $\Omega_k$.