ii. (40 points) Find all positive integers $n$ such that
$$
\sin \frac{2\left[\frac{n+1}{4}\right] \pi}{n} \geqslant \frac{n-1}{n} .
$$
When $n=1$, $\sin \frac{2\left[\frac{n+1}{4}\right] \pi}{n}=0=\frac{n-1}{n}$.
When $n=2$, $\sin \frac{2\left[\frac{n+1}{4}\right] \pi}{n}=0\frac{2}{3}=\frac{n-1}{n} \text {. }$

When $n=4$,
$$
\sin \frac{2\left[\frac{n+1}{4}\right] \pi}{n}=\sin \frac{\pi}{2}=1>\frac{3}{4}=\frac{n-1}{n} \text {. }
$$

When $n=5$, since $\sin x$ is a concave function on $[0, \pi]$, by Jensen's inequality we have
$$
\begin{aligned}
& \sin \frac{2\left[\frac{n+1}{4}\right] \pi}{n}=\sin \frac{2 \pi}{5}=\sin \frac{4 \times \frac{\pi}{2}}{5} \\
= & \sin \frac{4 \times \frac{\pi}{2}+1 \times 0}{5} \\
> & \frac{1}{5}\left(4 \times \sin \frac{\pi}{2}+1 \times \sin 0\right)=\frac{4}{5}=\frac{n-1}{n} .
\end{aligned}
$$

When $n \geqslant 6$, if $n \equiv-1(\bmod 4)$, then let $n=4 k-1\left(k \in \mathbf{N}_{+}\right)$.
By Jensen's inequality we have
$$
\begin{array}{l}
\sin \frac{2\left[\frac{n+1}{4}\right] \pi}{n}=\sin \frac{2 k \pi}{4 k-1} \\
=\sin \frac{4 k \times \frac{\pi}{2}}{4 k-1}=\sin \frac{(4 k-2) \times \frac{\pi}{2}+1 \times \pi}{4 k-1} \\
\geqslant \frac{1}{4 k-1}\left[(4 k-2) \sin \frac{\pi}{2}+1 \times \sin \pi\right] \\
=\frac{4 k-2}{4 k-1}=\frac{n-1}{n} .
\end{array}
$$

If $n \neq-1(\bmod 4)$, then
$$
\frac{n-3}{4}\frac{n-1}{n} .
\end{array}
$$

In summary, the positive integer $n$ is all positive integers except 2.