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What irrational expressions $A$ with radicals can be expressed as trigonometric rational fraction (not a series) with only rational multiplies of $\pi$.

Example:

$ \frac{1}{\sqrt5} = \frac{\sin\frac{\pi}{10}}{1- \sin\frac{\pi}{10}}$

Is there a general algorithm for such numbers to find such expressions?

UPD: Niven's theorem on Wiki [1] didn't help me to figure it out.

[1] https://en.wikipedia.org/wiki/Niven%27s_theorem

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    $\begingroup$ You realize that your question is too broad? What do you mean by finite trigonometric expressions? Which arguments are you allowed to use in the trigonometric functions? What coefficients can you put before the trigonometric functions? Etc. $\endgroup$
    – Beni Bogosel
    Commented Jul 27, 2020 at 18:49
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    $\begingroup$ Depending on how exactly you formalize this, a reasonably complete answer might be that those are precisely the numbers which are contained in some cyclotomic extension of $\mathbb Q$, or equivalently (by Kronecker-Weber theorem) in some abelian extension. $\endgroup$
    – Wojowu
    Commented Jul 27, 2020 at 19:30
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    $\begingroup$ Presumably you don't want something like $x = \tan(\arctan(x))$? $\endgroup$
    – Robert Israel
    Commented Jul 28, 2020 at 2:22
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    $\begingroup$ So do you only allow trigonometric functions of rational multiples of $\pi$? $\endgroup$
    – Robert Israel
    Commented Jul 29, 2020 at 14:43
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    $\begingroup$ @Matt F: Since $\sqrt{17}$ is an algebraic integer with minimal polynomial $x^2-17$ having Galois group of order $2$, thus abelian, the Kronecker-Weber theorem says it is. In fact, if $w$ is a primitive $17$'th root of unity, $\sqrt{17} = \pm (1+2{w}^{3}+2{w}^{5}+2{w}^{6}+2{w}^{7}+2{w}^{10}+2{w}^{11}+2{w}^{12}+2{w}^{14})$. $\endgroup$
    – Robert Israel
    Commented Jul 31, 2020 at 3:28

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