Product Of Sines

Geometry Level 4

$\large\sin\dfrac{\pi}{7}\sin\dfrac{2\pi}{7}\sin\dfrac{3\pi}{7}\sin\dfrac{\pi}{14}\sin\dfrac{3\pi}{14}\sin\dfrac{5\pi}{14}$

If the value of the above product can be written in the form of $\dfrac{\sqrt {a}}{b}$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

The answer is 71.

1 solution

Otto Bretscher
Apr 12, 2016

$\sin\left(\frac{k\pi}{14}\right)$ for $k=1..6$ are the positive roots of $\frac{U_{13}(x)}{x}=2^{13}x^{13}+...+14$ , where $U_{13}(x)$ is a Chebyshev polynomial of the second kind. The product of all roots is $\frac{14}{2^{13}}$ , by Viète, and the product of the positive roots is $\sqrt{\frac{14}{2^{13}}}=\frac{\sqrt{7}}{64}$ . The answer is $\boxed{71}$

Kind of surprised that you didn't you roots of unity. Still, great solution!

A Former Brilliant Member - 5 years, 2 months ago

We have done this problem with roots of unity many times, so I chose another approach for the sake of variety. See @Kartik Sharma ' s comment here , for example.

Roots of unity and Chebyshev polynomials are two of my favourite concepts in math.

Otto Bretscher - 5 years, 2 months ago

I've seen that in many of your problems... ;-)

A Former Brilliant Member - 5 years, 2 months ago

We've been taught that formula (with it's proof) in class!

A Former Brilliant Member - 5 years, 2 months ago

@A Former Brilliant Member – That's great! I don't think too many students "in the West" know that formula, even at the élite schools.

Otto Bretscher - 5 years, 2 months ago

Not able to understand please explain sir how that formula is used

DIVYA JAMPALA - 5 years, 2 months ago

Which part(s) do you need me to explain? Many of the properties of Chebyshev polynomials are explained here . I use those properties freely in my solution.

Otto Bretscher - 5 years, 2 months ago

Product Of Sines

The answer is 71.

1 solution

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