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Math Help - cos72 without calculator

  1. #1
    MHF Contributor Amer's Avatar
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    cos72 without calculator

    how I can find cos(72) without calculator


    I know that
    cos(72^o)=\frac{\sqrt{5}-1}{4}

    but how ??
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  2. #2
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    Quote Originally Posted by Amer View Post
    how I can find cos(72) without calculator


    I know that
    cos(72^o)=\frac{\sqrt{5}-1}{4}

    but how ??
    Trigonometry Angles--Pi/5 -- from Wolfram MathWorld
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  3. #3
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    Quote Originally Posted by Amer View Post
    how I can find cos(72) without calculator


    I know that
    cos(72^o)=\frac{\sqrt{5}-1}{4}

    but how ??
    Convert to radians:

    72\text{ deg} \cdot \frac{\pi \text{ rad}}{180\text{ deg}}= \frac{2\pi}{5}\text{ rad}.

    Now by double angle, \cos \left(\frac{2\pi}{5}\right)=2\cos^2\left(\frac{\pi  }{5}\right) - 1.

    Find \cos\left(\frac{\pi}{5}\right) either by using a table, or by de Moivre's Formula.
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  4. #4
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    ... or whistleralley.com/polyhedra/pentagon.htm for a nice account of the geometry of a regular pentagon and its connection with the golden ratio.
    Last edited by Opalg; July 31st 2009 at 12:24 PM. Reason: Fixed link (thanks skeeter)
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  5. #5
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    skeeter's Avatar
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    Quote Originally Posted by Opalg View Post
    ... or http://whistleralley.com/polyhedra/pentagon.htm for a nice account of the geometry of a regular pentagon and its connection with the golden ratio.
    Pentagon

    ... fixed link.
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  6. #6
    Senior Member pacman's Avatar
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    Working out in degrees, we have
    Let x =72 degrees, multiply both sides by 5;
    5x = 360
    2x + 3x = 360
    2x = 360-3x

    Then, taking cos of both side
    cos 2x = cos(360 -3x)
    cos 2x = cos 3x

    Using an identity for double angle and triple angle, equating them we have
    2cos^2 x -1= 4cos^3 x - 3 cos x
    4cos^3 x - 2cos^2 x - 3 cos x +1=0
    4cos^3 x - 4cos^2 x + 2 cos^2 x - 2 cos x - cos x +1=0
    4cos^2x(cos x -1) +2cos x( cos x -1) -1( cos x - 1) =0
    (cos x - 1) (4cos^x +2 cos x - 1) =0

    Either cos x =0 means x=90 degree
    Solve (4cos^x +2 cos x - 1) =0

    cos x = [-2 +(plus or minus){sq rt (4+16)] /8
    cos x = [-2 +(plus or minus){sq rt (20)] /8
    cos x = [-2 +2(plus or minus){sq rt (5)] /8
    cos x = [-1 +(plus or minus){sq rt (5)] /4

    Neglecting NEGATIVE values of cos x, because x lies in first quadrant
    Therefore cos 72 degrees = (sqrt(5) -1)/4

    cos 72 = 0.30901699437494742410229341718 . . . .
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  7. #7
    MHF Contributor Amer's Avatar
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    Thanks guys I get it
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