Evaluation of $\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}$

Question

Evaluation of $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} = $$

$\bf{My\; Try::}$ I have solved Using Direct formula::

$$\sin \frac{\pi}{n}\cdot \sin \frac{2\pi}{n}\cdot......\sin \frac{(n-1)\pi}{n} = \frac{n}{2^{n-1}}$$

Now Put $n=7\;,$ We get $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}\cdot \sin \frac{4\pi}{7}\cdot \sin \frac{5\pi}{7}\cdot \sin \frac{6\pi}{7}=\frac{7}{2^{7-1}}$$

So $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} =\frac{\sqrt{7}}{8}$$

Now my question is how can we solve it without using Direct Formula, Help me

Thanks

Answer 1

$$ \begin{align} \prod_{k=1}^3\sin\left(\frac{k\pi}7\right)^2 &=\prod_{k=1}^6\sin\left(\frac{k\pi}7\right)\tag{1}\\ &=-\frac1{64}\prod_{k=1}^6\left(e^{ik\pi/7}-e^{-ik\pi/7}\right)\tag{2}\\ &=\frac1{64}\prod_{k=1}^6\left(1-e^{-i2k\pi/7}\right)\tag{3}\\ &=\frac1{64}\lim_{z\to1}\prod_{k=1}^6\left(z-e^{-i2k\pi/7}\right)\tag{4}\\ &=\frac1{64}\lim_{z\to1}\frac{z^7-1}{z-1}\tag{5}\\[6pt] &=\frac7{64}\tag{6} \end{align} $$ Explanation:
$(1)$: $\sin(x)=\sin(\pi-x)$
$(2)$: $\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$
$(3)$: pull $\prod\limits_{k=1}^6e^{ik\pi/7}=-1$ out of the product
$(4)$: $1=\lim\limits_{z\to1}z$
$(5)$: $\prod\limits_{k=1}^6\left(z-e^{-i2k\pi/7}\right)=\frac{z^7-1}{z-1}$
$(6)$: evaluate limit

Therefore, taking the square root of $(6)$, we get $$ \prod_{k=1}^3\sin\left(\frac{k\pi}7\right)=\frac{\sqrt7}8\tag{7} $$

Answer 2

Using $2\sin a\sin b=\cos(a-b)-\cos(a+b)$ and $2\sin a\cos b=\sin(a+b)+\sin(a-b)$, write

$$\sin \frac{\pi}7\cdot\sin \frac{2\pi}7\cdot \sin \frac{3\pi}7 = \frac12\left(\cos\frac{\pi}7-\cos\frac{3\pi}7\right)\sin\frac{3\pi}7=\frac14\left(\sin\frac{4\pi}7+\sin\frac{2\pi}7-\sin\frac{\pi}7\right)\\=\frac14\left(\sin\frac{2\pi}7+\sin\frac{4\pi}7+\sin\frac{8\pi}7\right)$$

Then have a look at this question: Trigo Problem : Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$

Answer 3

This is a Gauss sum in disguise, but it can be computed through elementary geometry, without resorting to algebraic number theory. Let us consider a regular heptagon $P_1 P_2 P_3 P_4 P_5 P_6 P_7$ with center $O$ and take $A=P_1, B=P_5, C=P_6$. The angles of $ABC$ are $\frac{\pi}{7},\frac{2\pi}{7},\frac{4\pi}{7}$. By Euler's formula and the sine theorem

$$ [ABC] = \frac{abc}{4R} = 2R^2 \sin(A)\sin(B)\sin(C) $$ but we also have $$[ABC] = [AOB]+[BOC]-[AOC] = \frac{R^2}{2}\left(\sin\frac{4\pi}{7}+\sin\frac{2\pi}{7}-\sin\frac{6\pi}{7}\right) $$ hence the claim is equivalent to: $$ \sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7} = \frac{1}{2}\sqrt{7}. $$ and since the values of the sine function at $\frac{\pi}{7},\frac{2\pi}{7},\frac{4\pi}{7}$ are positive, the claim is also equivalent to: $$ \left(1-\cos\frac{2\pi}{7}\right)\left(1-\cos\frac{4\pi}{7}\right)\left(1-\cos\frac{8\pi}{7}\right)=\frac{7}{8}. $$ Chebyshev polynomials ensure that $\cos\frac{2\pi}{7},\cos\frac{4\pi}{7},\cos\frac{8\pi}{7}$ are algebraic conjugates over $\mathbb{Q}$, and they are the roots of $p(x)=8x^3+4x^2-4x-1$. By factoring such polynomial over $\mathbb{R}$ it follows that $$ p(x) = 8x^3+4x^2-4x-1 = 8\left(x-\cos\frac{2\pi}{7}\right)\left(x-\cos\frac{4\pi}{7}\right)\left(x-\cos\frac{8\pi}{7}\right) $$ and by evaluating at $x=1$ we get $$ 7 = p(1) = 8\left(1-\cos\frac{2\pi}{7}\right)\left(1-\cos\frac{4\pi}{7}\right)\left(1-\cos\frac{8\pi}{7}\right) $$ QED.

Answer 4

Hint;

Assume $$ \theta = (n\pi)/7$$

$$\implies 4\theta = n\pi -3\theta$$

Take the sine on both the sides An apply the required manipulations

You should get a cubic in $\sin^2 \theta$ Since we have defined out $\theta$, we are quite aware what the roots are going to be

So the numerical part in the end divided by the coefficient of $x^3$ gives you

$$ {{[\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}}]}^2 = 7/64$$

That should be it