Results 1 to 2 of 2

Math Help - prove

  1. #1
    Member
    Joined
    Nov 2006
    Posts
    152

    prove

    prove this
    Attached Thumbnails Attached Thumbnails prove-3390ffd7b21a77e4996623f1ddec5d29.png  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
    From
    Medgidia, Romania
    Posts
    1,252
    Thanks
    5
    \cos 36=\cos\frac{\pi}{5}=\cos\frac{2\pi}{10}=\sin\left  (\frac{\pi}{2}-\frac{2\pi}{10}\right)=\sin\frac{3\pi}{30}\Rightar  row

    \Rightarrow 1-2\sin ^2\frac{\pi}{10}=3\sin\frac{\pi}{10}-4\sin ^3\frac{\pi}{10}
    Let \sin\frac{\pi}{10}=t\Rightarrow 4t^3-2t^2-3t+1=0\Rightarrow t_1=1,t_{2,3}=\frac{-1\pm\sqrt{5}}{4}\Rightarrow \sin\frac{\pi}{10}=\frac{\sqrt{5}-1}{4}.
    Then \cos\frac{\pi}{5}=1-2\sin ^2\frac{\pi}{10}=\frac{\sqrt{5}+1}{4}

    Now, \cot 7.5=\cot\frac{15}{2}=\frac{1+\cos 15}{\sin 15}

    \sin 15=\sin (45-30)=\sin 45\cos 30-\sin 30-\cos 45=\frac{\sqrt{6}-\sqrt{2}}{4}

    \cos 15=\cos (45-30)=\frac{\sqrt{6}+\sqrt{2}}{4}

    Then \cot 7.5=\sqrt{6}+\sqrt{2}+\sqrt{3}+2\Rightarrow

    \Rightarrow 4\cos 36+\cot 7.5=\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{5}+\  sqrt{6}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove a/b and a/c then a/ (3b-7c)
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 23rd 2010, 05:20 PM
  2. prove,,,
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 1st 2010, 09:02 AM
  3. Prove |w + z| <= |w| +|z|
    Posted in the Algebra Forum
    Replies: 3
    Last Post: February 28th 2010, 05:44 AM
  4. Replies: 2
    Last Post: August 28th 2009, 02:59 AM
  5. How to prove that n^2 + n + 2 is even??
    Posted in the Algebra Forum
    Replies: 3
    Last Post: November 30th 2008, 01:24 PM

Search Tags


/mathhelpforum @mathhelpforum