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Math Help - Finding cos (pi / 5)

  1. #1
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    Finding cos (pi / 5)

    Hello. Sorry to be a nuisance to you guys but you've all been very helpful. This is going to be my last question.

    -----

    If:

    sin(3pi / 10) = (1 + squareroot 5) / 4

    Find the exact value of cos(pi / 5)

    -----

    This is the second part of a question. The first part mentioned something about sin(2pi / t) = cos t so I guess I have to somehow incorporate this identity. You don't have to solve it (although you could if you want to) but it would be great if you could provide me a starting point so I could work my way through. I was thinking that perhaps a unit circle diagram may help.

    Thankyou very much. Of course, all help is appreciated. (happy)
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  2. #2
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    Quote Originally Posted by sqleung View Post
    Hello. Sorry to be a nuisance to you guys but you've all been very helpful. This is going to be my last question.

    -----

    If:

    sin(3pi / 10) = (1 + squareroot 5) / 4

    Find the exact value of cos(pi / 5)

    [snip]
    1. Note that \frac{\pi}{2} - \frac{3 \pi}{10} = \frac{\pi}{5}.

    2. Complementary angle formula: \cos \left( \frac{\pi}{2} - A \right) = \sin A.

    So substitute A = \frac{3\pi}{10} into the above complemenatry angle formula.
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  3. #3
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    Thankyou very much! With your information, I managed to get this:

    Cos (pi / 5) = Cos((pi/2) - (3pi / 10))
    Cos (pi / 5) = Cos(pi/2)Cos(3pi/10) + Sin(pi/2)Sin(3pi/10)
    Cos (pi / 5) = Sin(3pi/10)
    Cos (pi / 5) = (1 + squareroot 5)/4

    Is that right?
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  4. #4
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    Quote Originally Posted by sqleung View Post
    Thankyou very much! With your information, I managed to get this:

    Cos (pi / 5) = Cos((pi/2) - (3pi / 10))
    Cos (pi / 5) = Cos(pi/2)Cos(3pi/10) + Sin(pi/2)Sin(3pi/10)
    Cos (pi / 5) = Sin(3pi/10)
    Cos (pi / 5) = (1 + squareroot 5)/4

    Is that right?
    Your answer is .......... correct

    (You can of course confirm it with a calculator by looking at appropriate decimal approximations of each side)
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  5. #5
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    Thanks!
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  6. #6
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    cos (pi/5)

    Quote Originally Posted by sqleung View Post
    Hello. Sorry to be a nuisance to you guys but you've all been very helpful. This is going to be my last question.

    -----

    If:

    sin(3pi / 10) = (1 + squareroot 5) / 4

    Find the exact value of cos(pi / 5)

    -----

    This is the second part of a question. The first part mentioned something about sin(2pi / t) = cos t so I guess I have to somehow incorporate this identity. You don't have to solve it (although you could if you want to) but it would be great if you could provide me a starting point so I could work my way through. I was thinking that perhaps a unit circle diagram may help.

    Thankyou very much. Of course, all help is appreciated. (happy)
    ********************************
    Direction to solution:
    Firstly, look at the roots of the equation z^5 + 1 = 0, where z is a complex number. This equation has 5 complex roots, where /z/=1 and arg(z) = (pi/5) + k*2(pi/5), with k element of .

    The equation can be written in two fashions:
    1) (z-1)*(z^2-2zcos(pi/5)+1)*(z^2-2zcos(3pi/5)+1)=0
    2) (z-1)*(z^4 - z^3 + z^2 - z+1) = 0

    These equations should be identical i.e. the coefficients are the same. This will yield two equations with cos(pi/5) and cos(3pi/5) the two unknowns.
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