My Math Forum Trigonometric Product

 Trigonometry Trigonometry Math Forum

 July 13th, 2017, 08:13 PM #1 Senior Member   Joined: Jul 2011 Posts: 398 Thanks: 15 Trigonometric Product If $\displaystyle \theta = \frac{2\pi}{2009},$ Then $\displaystyle \cos \theta \cdot \cos 2\theta\cdot \cos 3 \theta \cdot\cdots \cos 1004 \theta$ is
 July 16th, 2017, 12:51 PM #2 Global Moderator   Joined: Dec 2006 Posts: 18,156 Thanks: 1422 1/2^1004
 July 16th, 2017, 01:49 PM #3 Senior Member   Joined: Oct 2009 Posts: 142 Thanks: 60 This can be easily solved using complex numbers, let me start by giving a hint: $$z^n - 1 = \prod_{\nu = 1}^n (z - e^{2\pi\nu/n})$$ Try to rewrite the right hand side to the product of cosines. Thanks from greg1313
 July 17th, 2017, 03:10 AM #4 Global Moderator   Joined: Dec 2006 Posts: 18,156 Thanks: 1422 If $\theta = 2\pi/9$, using cos(A) ≡ sin(2A)/(2sin(A)) for non-zero A, $\cos(\theta)\cos(2\theta) \cos(3\theta)\cos(4\theta) = \dfrac{\sin(4\pi/9)\sin(8\pi/9)\sin(12\pi/9)\sin(16\pi/9)}{16\sin(2\pi/9)\sin(4\pi/9)\sin(6\pi/9)\sin(8\pi/9)} = 1/16$. The same method applies to the original problem. Thanks from greg1313

 Tags product, trigonometric

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post srecko Linear Algebra 1 October 27th, 2016 12:25 PM panky Trigonometry 6 December 28th, 2011 11:57 AM otaniyul Linear Algebra 0 October 30th, 2009 07:40 PM ^e^ Real Analysis 6 May 6th, 2007 05:27 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top