The following amazing identity can be checked directly by hand. $$\sin\left(\frac{\pi}{30}\right)\sin\left(\frac{7\pi}{30}\right)\sin\left(\frac{11\pi}{30}\right)\sin\left(\frac{13\pi}{30}\right)\sin\left(\frac{17\pi}{30}\right)\sin\left(\frac{19\pi}{30}\right)\sin\left(\frac{23\pi}{30}\right)\sin\left(\frac{29\pi}{30}\right) = \frac{1}{2^8} $$ This makes me wonder:
Question. Can we find arbitrarily long sequences of prime numbers $p_1<p_2<\cdots<p_k$ such that the product $$\sin\left(\frac{\pi}{n}\right)\sin\left(\frac{p_1\pi}{n}\right)\sin\left(\frac{p_2\pi}{n}\right)\cdots\sin\left(\frac{p_k\pi}{n}\right)$$ is rational for some integer $n>p_k$.
Other examples are: $$\begin{align} \sin\left(\frac{\pi}{4}\right)\sin\left(\frac{3\pi}{4}\right) &= \frac{1}{2} \\ \sin\left(\frac{\pi}{6}\right)\sin\left(\frac{3\pi}{6}\right)\sin\left(\frac{5\pi}{6}\right) &= \frac{1}{2^2} \\ \sin\left(\frac{\pi}{8}\right)\sin\left(\frac{3\pi}{8}\right)\sin\left(\frac{5\pi}{8}\right)\sin\left(\frac{7\pi}{8}\right) &= \frac{1}{2^3} \\ \sin\left(\frac{\pi}{12}\right)\sin\left(\frac{2\pi}{12}\right)\sin\left(\frac{5\pi}{12}\right)\sin\left(\frac{7\pi}{12}\right)\sin\left(\frac{11\pi}{12}\right) &= \frac{1}{2^5} \\ \sin\left(\frac{\pi}{18}\right)\sin\left(\frac{5\pi}{18}\right)\sin\left(\frac{7\pi}{18}\right)\sin\left(\frac{11\pi}{18}\right)\sin\left(\frac{13\pi}{18}\right)\sin\left(\frac{17\pi}{18}\right) &= \frac{1}{2^6} \\ \sin\left(\frac{\pi}{18}\right)\sin\left(\frac{3\pi}{18}\right)\sin\left(\frac{5\pi}{18}\right)\sin\left(\frac{7\pi}{18}\right)\sin\left(\frac{11\pi}{18}\right)\sin\left(\frac{13\pi}{18}\right)\sin\left(\frac{17\pi}{18}\right) &= \frac{1}{2^7} \\ \sin\left(\frac{\pi}{30}\right)\sin\left(\frac{7\pi}{30}\right)\sin\left(\frac{11\pi}{30}\right)\sin\left(\frac{13\pi}{30}\right)\sin\left(\frac{17\pi}{30}\right)\sin\left(\frac{19\pi}{30}\right)\sin\left(\frac{23\pi}{30}\right)\sin\left(\frac{29\pi}{30}\right) &= \frac{1}{2^8} \\ \sin\left(\frac{\pi}{30}\right)\sin\left(\frac{5\pi}{30}\right)\sin\left(\frac{7\pi}{30}\right)\sin\left(\frac{11\pi}{30}\right)\sin\left(\frac{13\pi}{30}\right)\sin\left(\frac{17\pi}{30}\right)\sin\left(\frac{19\pi}{30}\right)\sin\left(\frac{23\pi}{30}\right)\sin\left(\frac{29\pi}{30}\right) &= \frac{1}{2^9} \end{align} $$ The longest I found is $$\sin\left(\frac{\pi}{38}\right)\sin\left(\frac{3\pi}{38}\right)\sin\left(\frac{5\pi}{38}\right)\sin\left(\frac{7\pi}{38}\right)\sin\left(\frac{11\pi}{38}\right)\sin\left(\frac{13\pi}{38}\right)\sin\left(\frac{17\pi}{38}\right)\sin\left(\frac{19\pi}{38}\right)\sin\left(\frac{23\pi}{38}\right)\sin\left(\frac{29\pi}{38}\right) = \frac{1}{2^9}$$ Note that it breaks the pattern of $n=p_k+1$. Note also that not all examples have consecutive primes.