We have the following identities:
$\sin(\frac{\pi}{1})=0$
$\sin(\frac{\pi}{2})=1$
$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{\sqrt{4}}$
$\sin(\frac{\pi}{4})=\frac{1}{2}$
$\sin(\frac{\pi}{5})=\frac{\sqrt{5-\sqrt{5}}}{\sqrt{8}}$
On the other hand it is known that for integer $n$ , $\sin(\frac{\pi}{n}) $ can not always be expressed by real radicals.
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With expressed by real radicals, I mean expressible with sums and products and integers and n'th roots of such positive expressions. So no stuff like $(-1)^{1/7}$ or $i$ allowed.
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And it is known that for integer $n$ , $\sin(\frac{\pi}{n}) $ can not always be expressed by $\sin(\frac{\pi}{q_i}) $ for a collection of $i$ integers $q_i$ relatively prime to $n$.
So combining those ideas, I wonder
When is $\sin(\frac{\pi}{n}) $ not expressible by real radicals and $\sin(\frac{\pi}{q_i}) $ ?
I assume the factorization of $n$ matters and it is only possible when it is possible for all primes that divide $n$
Im not completely sure though but this lead me to consider the main question :
For a prime $p$ :
When is $\sin(\frac{\pi}{p}) $ not expressible by real radicals and $\sin(\frac{\pi}{q_i}) $ ?
See also :
Algebraic numbers expressible in terms of real-valued radicals