User Name Remember Me? Password

 Trigonometry Trigonometry Math Forum

 August 11th, 2010, 05:00 AM #1 Newbie   Joined: Aug 2010 Posts: 7 Thanks: 0 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. August 11th, 2010, 07:55 AM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 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. August 11th, 2010, 07:59 AM #3 Global Moderator   Joined: Nov 2009 From: Northwest Arkansas Posts: 2,767 Thanks: 5 Re: Product of sine FTOC or FTOA ?? I'm either really stupid, or...  August 11th, 2010, 08:01 AM #4 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Product of sine No, you're right. Pesky Fundamental Theorems cropping up everywhere and confusing me.  August 11th, 2010, 08:15 AM #5 Global Moderator   Joined: Nov 2009 From: Northwest Arkansas Posts: 2,767 Thanks: 5 Re: Product of sine And I assume you mean "algebra" and not "arithmetic"!              August 11th, 2010, 08:24 AM #6 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Product of sine Oh, shut up  August 11th, 2010, 07:42 PM #7 Newbie   Joined: Aug 2010 Posts: 7 Thanks: 0 Re: Product of sine Get it! Thanks mattpi!!  Tags product, sine Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post nowayyou Trigonometry 3 January 28th, 2013 06:54 PM johnny Calculus 1 March 12th, 2011 02:29 AM moja_a Trigonometry 8 February 27th, 2011 09:33 PM otaniyul Linear Algebra 0 October 30th, 2009 06:40 PM ^e^ Real Analysis 6 May 6th, 2007 04:27 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top      