Is there a way to determine which angles, $3\theta$, are trisectable into $\theta$? Even when the angle $3\theta$ isn't constructible to begin with?

Question

Is there a way to determine which angles, $3\theta$, are trisectable into $\theta$? Even when the angle $3\theta$ isn't itself constructible to begin with?

Answer 1

This is taken directly from Wikipedia.

An angle $\theta$ can be trisected iff $4t^3-3t-\cos\theta$ is reducible over $\mathbb Q(\cos\theta)$.

For example, $3\pi/7$ is trisectible because, letting $c=\cos3\pi/7$, $$4t^3-3t-c=4(t+2c^2-1)(t^2+(1-2c^2)t+c/2-c^2)$$ Here $8c^3-4c^2-4c+1=0$.