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First question: For which angles $x$ is $\sin(x)$ a real number that can be expressed using only integers, addition, subtraction, multiplication, division and the extraction of $n$th roots? (With "$n$th roots" meaning real-valued $n$th roots of positive numbers). Let $A$ be the set of numbers in $[-1; 1]$ expressible in this way.

Unfortunately there is a lot of confusing information about this topic floating around. For example, in this question, the accepted answer claims:

But it is known, for example, that there is no expression for $\sin(\pi/9)$ that starts from the integers, and uses only the ordinary operations of arithmetic and roots in which every component is real.

This seems to be false. The trigonometric formula $\cos(3x) = 4 \cos^3(x) - 3\cos(x)$ should yield the value of $cos(20^\circ)$ as the solution of a cubic equation.

At other places there are claims that the $n$th cyclotomic polynomial has an abelian Galois-group and is therefore solvable. Does this imply that $\sin(2\pi/n)$ is in $A$ for every positive integer $n$? (Sorry if this is a stupid question, I don't understand algebra.)

If so, $\sin(\pi x)$ would also be in $A$ for every rational $x$ (and $\sin(\pi x)$ not in $A$ would imply that $x$ must be irrational).

If this is true, I have a second question: Does the inverse also hold? Is every number in $A$ of the form $\sin(\pi x)$ for a rational $x$?

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The trigonometric formula $\cos(3x)=4\cos^3(x)−3\cos(x)$ should yield the value of $\cos(20^\circ)$ as the solution of a cubic equation.

True. Unfortunately it is the irreducible case, and the in the process requires taking square root of a negative number:

$$\cos(20^\circ) = \sqrt[3]{\frac{1 + \sqrt{-3}}{16}} + \sqrt[3]\frac{1 - \sqrt{-3}}{16}$$

The answer you mentioned emphasizes that every component is real.

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