Is the value of $\sin(\frac{\pi}{n})$ expressible by radicals?

Question

We have the followings:

$\sin(\frac{\pi}{1})=\frac{\sqrt{0}}{\sqrt{1}}$

$\sin(\frac{\pi}{2})=\frac{\sqrt{1}}{\sqrt{1}}$

$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{\sqrt{4}}$

$\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{\sqrt{4}}$

$\sin(\frac{\pi}{5})=\frac{\sqrt{5-\sqrt{5}}}{\sqrt{8}}$

Question: Is the value of $\sin({\frac{\pi}{n}})$ expressible by fractions, radicals and natural numbers for each given $n$? If not, for which $n$ can we prove this non-expressibility?

Answer 1

I don't think it works for $n=90$ because that would essentially mean sin2° and that it as far as I know not expressible in terms of radical terms

Answer 2

Cyclotomic polynomials are solvable, so their imaginary parts are expressible in radicals. A separate issue is whether the values are constructible -- i.e., whether all the radical sign are square roots.

Answer 3

As was alluded to in comments this relates to the nth roots of unity. First of all you distinguish between 'primitive' and non-primitive. Then you look if those primitive roots are expressible in radicals and so on. The case where they are not happens for example with the Casus irreducibilis, which sometimes (!) happens in trisecting the angle (but never in bisecting the angle). Long story short it turns out for example that for $0<=k<=2^n$ all $\sin(2\pi k/2^n)$ and $\cos(2\pi k/2^n)$ are expressible in radicals!

Answer 4

You'll find an excellent start here: http://www.efnet-math.org/Meta/sine1.htm

Answer 5

If $f(x) = 64x^6 - 112x^4 + 56x^2 - 7$ then $f(\sin(\pi/7)) = 0$ , but I think this was known in some form to Kepler.

Which suggests that we look for the minimal polynomial which has $\sin(\pi/n)$ as a root.

Here is a reference:Scott Beslin and Valerio de Angelis, The minimal polynomials of sin(2 pi/p) and cos(2 pi/n), Math. Mag., 77 (2004), 146-149.