Please see the picture for an illustration.
It is given, that $\theta = \frac{\pi}{2N}$ for $N \geq 2$ a natural number and the height from the bottom of the triangle to the top is $\epsilon$. I need to estimate the length of the side $A$ in terms of $N$ and $\epsilon$, as far as I know the estimate $A \leq \frac{8\epsilon}{N} $ should hold but I am really having issues showing it. In particular having the estimate involving $N$, when using simple trigonometric $\sin$/$\cos$/$\tan$ identities.
Any help would be greatly appreciated!
Let $\Delta PQR$ be our triangle, $QR=A$, $\measuredangle QPR=\theta$, $PK$ be an altitude of $\Delta PQR$ such that ray $PQ$ placed between rays $PK$ and $PR$. By the given $PK=\epsilon$.
Now, let $PL$ be a ray such that $PL\perp PK$ and $\{L,Q,R\}$ placed at the same side related to the line $PK$.
We'll rote $\angle QPR$ around point $P$ such that ray $PR\rightarrow$ ray $PL$ and $QR||PL$.
Easy to see that $A=QR\rightarrow+\infty$ for all $\epsilon>0$ and for all $N\geq2$.