1. ## 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?

2. Originally Posted by elmo
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?

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.