July 13th, 2017, 08:13 PM  #1 
Senior Member Joined: Jul 2011 Posts: 399 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: 18,593 Thanks: 1492 
1/2^1004

July 16th, 2017, 01:49 PM  #3 
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 84 
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. 
July 17th, 2017, 03:10 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,593 Thanks: 1492 
If $\theta = 2\pi/9$, using cos(A) ≡ sin(2A)/(2sin(A)) for nonzero 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. 

Tags 
product, trigonometric 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
combination of scalar product and vector product equation  srecko  Linear Algebra  1  October 27th, 2016 12:25 PM 
trigonometric value  panky  Trigonometry  6  December 28th, 2011 11:57 AM 
what is the difference? cartesian product tensor product etc  otaniyul  Linear Algebra  0  October 30th, 2009 07:40 PM 
Inner product product rule  ^e^  Real Analysis  6  May 6th, 2007 05:27 PM 