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Math Help - Trigonometric Identity

  1. #1
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    Trigonometric Identity

    Prove the following identity.
    cos2pi/7 +cos4pi/7 + cos8pi/7 = -1/2
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  2. #2
    Junior Member mathbyte's Avatar
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    Re: Trigonometric Identity

    We have
    cos(2pi/7) +cos(4pi/7) + cos(8pi/7) = -1/2

    The best way to prove this will be to simplify the left side. To do that we need to be able to try and collect like terms.

    There is a trigonometric identity that says:
    sin(x)cos(y) = (1/2)[(sin(x+y)+sin(x-y)]

    Let's multiply both sides of the equation by sin(pi/7):
    sin(pi/7)cos(2pi/7) + sin(pi/7)cos(4pi/7) + sin(pi/7)cos(6pi/7) = -1/2

    Now, we apply the identity above:
    (1/2)[sin(pi/7+2pi/7) + sin(pi/7-2pi/7)] + (1/2)[sin(pi/7+4pi/7) + sin(pi/7-4pi/7)]
    + (1/2)[sin(pi/7+8pi/7) + sin(pi/7-8pi/7)] = -(1/2)sin(pi/7)

    We can multiply both sides by 2 and simplify the terms a bit:
    [sin(3pi/7) + sin(-pi/7)] + [sin(5pi/7) + sin(-3pi/7)] + [sin(9pi/7) + sin(-7pi/7)] = -sin(pi/7)

    Further simplifying:
    sin(3pi/7) - sin(pi/7) + sin(5pi/7) - sin(3pi/7) + sin(9pi/7) - sin(7pi/7) = -sin(pi/7)

    Now, we can collect like terms:
    sin(5pi/7) + sin(9pi/7) - sin(7pi/7) = 0

    But sin(7pi/7) = 0:
    sin(5pi/7) + sin(9pi/7) = 0

    At this point you can probably inference whether the identity will be proved, or not.

    By the way, was the problem stated/copied correctly? I find myself wondering of the last term of the left half of the equation should be cos(6pi/7), not cos(8pi/7).

    Hope this helps!
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  3. #3
    MHF Contributor Amer's Avatar
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    Re: Trigonometric Identity

    sin(5pi/7) + sin(9pi/7) = 0 this is true, write 5pi/7 = pi - 2pi/7 , 9pi/7 = pi + 2pi/7 and use the identities

    sin(a+b), sin(a-b)
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  4. #4
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    Re: Trigonometric Identity

    Quote Originally Posted by ematrix View Post
    Prove the following identity.
    cos2pi/7 +cos4pi/7 + cos8pi/7 = -1/2
    Here's a picture to illustrate this problem. It shows seven spots equally spaced round the unit circle. The black spot is at (1,0). The x-coordinates of the red spots are \cos2\pi/7, \cos4\pi/7 and \cos8\pi/7, as also are the x-coordinates of the blue spots. The centre of mass of the seven spots is the origin. So the sum of the x-coordinates of the red spots plus the x-coordinates of the blue spots plus 1 (the x-coordinate of the black spot) is equal to 0. It follows that the sum of the x-coordinates of the red (or the blue) spots is 1/2.

    That is an "applied" approach to the problem. You probably want a pure mathematical proof. There are several possible methods for this. One is to look at solutions of the equation \cos 7x = 1. Another is to use complex numbers: if z=e^{2\pi i/7} then z^7=1. But I think that the applied approach is the one that gives the most genuine insight as to why the result is true.
    Attached Thumbnails Attached Thumbnails Trigonometric Identity-spots.png  
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