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Thread: constructible angle

  1. #1
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    constructible angle

    cosθ=3/7, θ is an acute angle.
    prove θ cannot be trisected with straightedge and compass?




    my approach: angle θ can't be constructed with straightedge and compass if cosθ is transcendental, but cosθ=3/7 is algebraic and so it is not transcendental?

    Please help me with this question, thanks.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by elmo View Post
    cosθ=3/7, θ is an acute angle.
    prove θ cannot be trisected with straightedge and compass?




    my approach: angle θ can't be constructed with straightedge and compass if cosθ is transcendental, but cosθ=3/7 is algebraic and so it is not transcendental?

    Please help me with this question, thanks.
    The question you're asking is if the angle  \frac\theta3 is constructible.

    The angle  \frac\theta3 is constructible  \iff \cos\left(\frac\theta3\right) and  \sin\left(\frac\theta3\right) are both constructible numbers.

    Let's compute  \cos\left(\frac\theta3\right) by the triple angle formula:

     \cos(\theta) = 4\cos\left(\frac\theta3\right)^3-3\cos\left(\frac\theta3\right)

    So we see  \cos\left(\frac\theta3\right) is a root of  f(x)=28x^3-21x-3 .

     f(x) has no linear factors and thus is irreducible. Hence  f(x) =  m_{\cos\left(\frac\theta3\right),\mathbb{Q}}(x) .

    Since a number  \alpha is constructible  \iff \text{deg}(m_{\alpha,\mathbb{Q}}(x)) = 2^n , we see that  \cos\left(\frac\theta3\right) is not constructible  \implies the angle  \frac\theta3 isn't either.
    Last edited by chiph588@; Apr 2nd 2010 at 08:48 AM.
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