2
$\begingroup$

From the article Products of Sines, we have $\sin 15^\circ\sin75^\circ=\sin 18^\circ\sin54^\circ=\frac{1}{4}$. We can rewrite this as $\sin \frac{\pi}{12}\sin\frac{5\pi}{12}=\sin \frac{\pi}{10}\sin\frac{3\pi}{10}=\frac{1}{4}$. Is there any good method to get $x,y$ such that $\sin x\sin y=\frac{1}{4}$ or more generally to get $x_i$ such that $\prod_{i=1}^n\sin x_i=k$ where $k$ is a rational number?

  • 2
    I wrote a paper in which I found all rational products of three and of four sines of rational angles (the case of two had already been done). Rational products of sines of rational angles, Aequationes Math. 45 (1993) 70-82, MR 93m:11140. Maybe this link works: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002039893&IDDOC=1780382012-10-24

0 Answers 0