7、The smallest positive integer $n$ that satisfies $n \sin 1 > 5 \cos 1 + 1$ is

保留了源文本的换行和格式，以下是翻译结果：

7、The smallest positive integer $n$ that satisfies $n \sin 1 > 5 \cos 1 + 1$ is
7.5

Analysis: Since $n \sin \frac{\pi}{3}>n \sin 1>5 \cos 1+1>5 \cos \frac{\pi}{3}+1$, hence $n \geq 5$; it is sufficient to prove: $5 \sin 1>5 \cos 1+1$, which means it is sufficient to prove $\sin \left(1-\frac{\pi}{4}\right)>\frac{\sqrt{2}}{10}$.

Since $y=\sin x$ is a concave function on $\left(0, \frac{\pi}{4}\right)$, hence $\sin x>\frac{2 \sqrt{2}}{\pi} x$,
and $0<1-\frac{\pi}{4}<\frac{\pi}{4}$, thus $\sin \left(1-\frac{\pi}{4}\right)>\frac{2 \sqrt{2}}{\pi}\left(1-\frac{\pi}{4}\right)>\frac{\sqrt{2}}{10}$.