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August 11th, 2010, 05:00 AM   #1
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Product of sine

Hi! I have been thinking about this problem for a rather long time, it is taken from a textbook:
Prove that
$\displaystyle \displaystyle \prod_{k=1}^{13} \sin \frac{k\pi}{27}=\frac{3\sqrt{3}}{2^{13}}$.
Any help is appreciated!

Last edited by skipjack; February 28th, 2018 at 02:38 PM.
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August 11th, 2010, 07:55 AM   #2
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Re: Product of sine

The solution to this is a rather neat one, so I don't want to give it away straight away. Instead, I'll give you a hint, and see how far you can get.

Let $\displaystyle n\in\mathbb{N}$ be odd, and consider the polynomial equation $\displaystyle x^n-1=0.$ We know that the $\displaystyle n$ roots of this equation are $\displaystyle 1,\omega_n,\omega_n^2,\ldots,\omega_n^{n-1}$ where $\displaystyle \omega_n=e^{2i\pi/n}.$ Therefore, by the FTOA we can write

$\displaystyle x^n-1=(x-1)(x-\omega_n)(x-\omega_n^2)\cdots(x-\omega_n^{n-1}),$

or, since $\displaystyle \omega_n^{n-k}=\omega_n^n\omega_n^{-k}=1\cdot\omega_n^{-k},$

$\displaystyle \begin{align}x^n-1&=(x-1)(x-\omega_n)(x-\omega_n^{-1})(x-\omega_n^2)(x-\omega_n^{-2})\cdots(x-\omega_n^{(n-1)/2})(x-\omega_n^{-(n-1)/2})\\
&=(x-1)[x^2-(\omega_n+\omega_n^{-1})x+1][x^2-(\omega_n^2+\omega_n^{-2})x+1]\cdots[x^2-(\omega_n^{(n-1)/2}+\omega_n^{-(n-1)/2})x+1].\end{align}$

Now divide both sides by $\displaystyle (x-1),$ evaluate the equation at $\displaystyle x=1$ (either take the limit on the left or express the fraction as a sum of powers of $x$) and write the RHS in terms of trigonometric functions. You will need to remember that $\displaystyle \cos\theta=(e^{i\theta}+e^{-i\theta})/2$ and that $\displaystyle \sin\theta=\sin(\pi-\theta).$

Last edited by skipjack; February 28th, 2018 at 02:41 PM.
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August 11th, 2010, 07:59 AM   #3
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Re: Product of sine

FTOC or FTOA ??
I'm either really stupid, or...
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August 11th, 2010, 08:01 AM   #4
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Re: Product of sine

No, you're right. Pesky Fundamental Theorems cropping up everywhere and confusing me.
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August 11th, 2010, 08:15 AM   #5
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Re: Product of sine

And I assume you mean "algebra" and not "arithmetic"!




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August 11th, 2010, 08:24 AM   #6
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Re: Product of sine

Oh, shut up
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August 11th, 2010, 07:42 PM   #7
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Re: Product of sine

Get it! Thanks mattpi!!
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