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Thread: Trig product

  1. #1
    Senior Member DivideBy0's Avatar
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    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:

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

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

    Thanks
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  2. #2
    MHF Contributor red_dog's Avatar
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    $\displaystyle P=\cos\frac{\pi}{11}\cos\frac{2\pi}{11}\cos\frac{3 \pi}{11}\cos\frac{4\pi}{11}\cos\frac{5\pi}{11}=$

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

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

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

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

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

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

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

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

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

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

    $\displaystyle \Rightarrow P=\frac{1}{32}$
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