TITLE: Finding $ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$
QUESTION [20 upvotes]: I would appreciate if somebody could help me with the following problem. How can we find the product 
$$ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$$

REPLY [6 votes]: I have an even more precise bound: the product is less than
$$(999)! \left(\frac{\pi}{1999}\right)^{999}$$
Use Stirling approximation to get that this in turn is less than
$$ \sqrt{999 e} \left ( \frac{999 \pi}{1999 e}\right )^{999} \approx 4 \times 10^{-237}$$