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Math Help - Can I express cos(pi/7) with radicals?

  1. #1
    Newbie Maxim's Avatar
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    Can I express cos(pi/7) with radicals?

    I'm trying to express cos(pi/7) with radicals - that is, basic operations (addition, division, etc) and square and cube roots (or even higher roots if necessary).

    I got this far:

    If s = 84\sqrt{3} \cdot i - 28 and c = \sqrt[3]{s}, then x = \frac{2+c}{12} + \frac{7}{3c} = \cos(\pi/7)

    Although this seems a complex expression, the imaginary part of x is actually zero, so x is in fact a real number (obviously, since cos(pi/7) is real). But I can't extract the real part as a "really real" radical expression (without i, that is).
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  2. #2
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    Quote Originally Posted by Maxim View Post
    I'm trying to express cos(pi/7) with radicals - that is, basic operations (addition, division, etc) and square and cube roots (or even higher roots if necessary).

    I got this far:

    If s = 84\sqrt{3} \cdot i - 28 and c = \sqrt[3]{s}, then x = \frac{2+c}{12} + \frac{7}{3c} = \cos(\pi/7)

    Although this seems a complex expression, the imaginary part of x is actually zero, so x is in fact a real number (obviously, since cos(pi/7) is real). But I can't extract the real part as a "really real" radical expression (without i, that is).
    Read this: Trigonometry Angles--Pi/7 -- from Wolfram MathWorld
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  3. #3
    Newbie Maxim's Avatar
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    Thanks. Still a bit confused though, the page says:
    Quote Originally Posted by Wolfram Mathworld
    The angles \pi\frac{m}{n} (with m,n integers) for which the trigonometric functions may be expressed in terms of finite root extraction of real numbers are limited to values of m which are precisely those which produce constructible polygons.
    Which is also what I read elsewhere (except for the typo - that last condition should be "values of n")

    But further down, it says:
    In general, any trigonometric function can be expressed in radicals for arguments of the form \pi\frac{m}{n} (...)
    From the example, I figured that \cos(\pi/7) = -\frac{1}{2} \cdot (-1)^{6/7} \cdot \left(1+(-1)^{2/7}\right), doesn't that fit the description "in terms of finite root extraction of real numbers"?
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