Find the value of $\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\pi}{14}$

Question

Find the value of $$\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\pi}{14}$$

$$\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\pi}{14}$$ $$=\sin\frac{2\pi}{14}\sin\frac{4\pi}{14}\sin\frac{6\pi}{14}\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\pi}{14}$$ $$=\sin\frac{\pi}{14}\sin\frac{2\pi}{14}\sin\frac{3\pi}{14}\sin\frac{4\pi}{14}\sin\frac{5\pi}{14}\sin\frac{6\pi}{14}$$

How can I compute $$\prod\limits_{r=1}^{n}\sin\frac{r\pi}{c}$$

Answer 1

It can be seen as an instance of the well-known identity: $$ \prod_{k=1}^{n-1}\sin\left(\frac{k\pi}{n}\right)=\frac{2n}{2^n} \tag{1} $$ that for $n=14$ gives: $$ \left[\prod_{k=1}^{6}\sin\left(\frac{k\pi}{14}\right)\right]^2 = \frac{14}{2^{13}} \tag{2}$$ from which: $$ \prod_{k=1}^{6}\sin\left(\frac{k\pi}{14}\right) = \color{red}{\frac{\sqrt{7}}{64}}.\tag{3} $$

Answer 2

By the basic properties of Chebyshev polynomials the numbers $x_k=\sin(k\pi/14)$, $k=-6,-5,\ldots,5,6, k\neq0$ are the twelve zeros of the polynomial $$ \frac{U_{13}(x)}x=8192 x^{12}-24576 x^{10}+28160 x^8-15360 x^6+4032 x^4-448 x^2+14. $$

By the usual relations (aka Vieta relations) of the zeros of a polynomial and its coefficients it follows that $$ A=\prod_{k=-6..6, k\neq0}x_k=\frac{14}{8192}=\frac{7}{2^{12}}. $$

Because sine is known to be an odd function, the products $N=\prod_{k=-6}^{-1}x_k$ and $P=\prod_{k=1}^6x_k$ are equal. You are asking about the product $P$. It is clearly a positive number, so from the equation $$A=NP=P^2=\frac{7}{2^{12}} $$ we can solve $$ P=\frac{\sqrt7}{2^6}. $$

The method generalizes for other products of sines as long as you take the product of ALL the numbers $\sin (k\pi/(2N))$, $k=1,2,\ldots,N-1$.

Answer 3

Use $\sin\left(\dfrac\pi2-A\right)=\cos A$ and $\sin2B=2\sin B\cos B$

to find the product $$=\dfrac{\sin2t\sin4t\sin6t}8$$ where $7t=\pi$

Now use Method to find $\sin (2\pi/7)$, to find

$$\sin2t\sin4t\sin6t=\sqrt{\dfrac7{64}}$$