Results 1 to 2 of 2

Math Help - Trig product

  1. #1
    Senior Member DivideBy0's Avatar
    Joined
    Mar 2007
    From
    Melbourne, Australia
    Posts
    432

    Trig product

    Wow I haven't been on here for a long, long time. I'm sorry for leaving so abruptly... I've been really busy lately and I find it very easy to procrastinate on forums. Oh well, I'm back now, hi.

    Find the product:

    \cos{\frac{\pi}{11}}\cos{\frac{2\pi}{11}}\cos{\fra  c{3\pi}{11}}\cos{\frac{4\pi}{11}}\cos{\frac{5\pi}{  11}}

    I have tried using several methods but I haven't gotten anywhere

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
    From
    Medgidia, Romania
    Posts
    1,245
    Thanks
    1
    P=\cos\frac{\pi}{11}\cos\frac{2\pi}{11}\cos\frac{3  \pi}{11}\cos\frac{4\pi}{11}\cos\frac{5\pi}{11}=

    =\cos\frac{\pi}{11}\cos\frac{2\pi}{11}\cos\frac{4\  pi}{11}\cos\frac{8\pi}{11}\cos\frac{6\pi}{11}

    I used \cos\frac{3\pi}{11}=\cos\left(\pi-\frac{8\pi}{11}\right)=-\cos\frac{8\pi}{11}, \ \cos\frac{5\pi}{11}=\cos\left(\pi-\frac{6\pi}{11}\right)=-\cos\frac{6\pi}{11}

    Multiply both members by \sin\frac{\pi}{11}

    \sin\frac{\pi}{11}P=\frac{1}{2}\sin\frac{2\pi}{11}  \cos\frac{2\pi}{11}\cos\frac{4\pi}{11}\cos\frac{8\  pi}{11}\cos\frac{6\pi}{11}=

    =\frac{1}{4}\sin\frac{4\pi}{11}\cos\frac{4\pi}{11}  \cos\frac{8\pi}{11}\cos\frac{6\pi}{11}=

    =\frac{1}{8}\sin\frac{8\pi}{11}\cos\frac{8\pi}{11}  \cos\frac{6\pi}{11}=

    =\frac{1}{16}\sin\frac{16\pi}{11}\cos\frac{6\pi}{1  1}=

    =-\frac{1}{16}\sin\frac{5\pi}{11}\cos\frac{6\pi}{11}  =

    =-\frac{1}{32}\left(\sin\left(\frac{5\pi}{11}+\frac{  6\pi}{11}\right)+\sin\left(\frac{5\pi}{11}-\frac{6\pi}{11}\right)\right)=

    =-\frac{1}{32}\left(\sin\pi-\sin\frac{\pi}{11}\right)=\frac{1}{32}\sin\frac{\p  i}{11}

    \Rightarrow P=\frac{1}{32}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. product of inverse trig function
    Posted in the Calculus Forum
    Replies: 4
    Last Post: November 12th 2011, 08:53 AM
  2. Replies: 6
    Last Post: September 7th 2010, 09:03 PM
  3. Integration of nasty trig product
    Posted in the Calculus Forum
    Replies: 5
    Last Post: December 5th 2009, 01:45 PM
  4. multivariable differential for inner product(scalar product)?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 23rd 2009, 05:40 PM
  5. Replies: 4
    Last Post: October 13th 2009, 09:28 AM

Search Tags


/mathhelpforum @mathhelpforum