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How many different tangent angles create reducible expressions like $\tan(60^\circ)=\sqrt3$? Please include obscure angles. Where would information like this be found? There are in infinite number of degree steps to be considered. How to address this problem?

Michael Hardy
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User3910
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    What precisely do you mean by reducible? –  Nov 16 '17 at 02:37
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    Do you mean something like this? – user222031 Nov 16 '17 at 02:38
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    If the trig angle can be expressed as a polynomial, then it is often solvable. Example as $\cos(\theta) = -\cos(2\theta)$ which is $\cos(\theta) + 2\cos(\theta)^2 - 1 = 0$ has a solution $\theta = 60 \text{ degrees}$, and solving for $\cos(\theta)$ gives you the radical value. – DanielV Nov 16 '17 at 02:42
  • @user222031 yes very much like that. Is that exhaustive? – User3910 Nov 16 '17 at 02:42
  • @DanielV that is https://en.wikipedia.org/wiki/Baker%27s_theorem – User3910 Nov 16 '17 at 02:45
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    @Dale Then, the answer is that, for infinitely (but countably) many angles $\theta \in \mathbb{Q}\pi \cap \left(-\frac{\pi}{2},+\frac{\pi}{2}\right)$, $\tan(\theta)$ can be expressed via radicals. However, if you require that $\theta$ is of the form $\frac{p}{q}\pi$ for some rational numbers $\frac{p}{q}\in\left(-\frac12,+\frac12\right)$ with fixed $q\in\mathbb{Z}_{>0}$, then I would suggest that you look at $q=2^r F_1F_2\cdots F_s$ where $r$ is a nonnegative integer and $F_1,\ldots,F_s$ are distinct Fermat primes. – Batominovski Nov 16 '17 at 02:46
  • @Batominovski I would accept that as the answer if you would kindly paste it there. Are you saying that Fermat prime numbers are directly related to the radicals found in the tangent function? Also, is there a reason you included 8 Fermat primes? – User3910 Nov 16 '17 at 02:52
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    It is known that the trigonometric functions of an integer number of degrees may be expressed by radicals if the number of degrees is divisible by 3: https://www.mapleprimes.com/posts/144188-Trigonometric-Functions-In-Radicals – cgiovanardi Nov 16 '17 at 20:55

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For infinitely (but countably) many angles $\theta\in\mathbb{Q}\pi\cap\left(-\frac\pi2,+\frac\pi2\right)$, the number $\tan(\theta)$ can be expressed via radicals in the sense of the link given by user222031 in a comment above. However, if you require that $\theta$ is of the form $\frac{p}{q}\pi$ for some rational numbers $\frac{p}{q}\in\left(−\frac12,+\frac12\right)$ with a fixed denominator $q\in\mathbb{Z}_{>0}$, then I would suggest that you look at $q=2^rf_1f_2\cdots f_s$, where $r$ and $s$ are nonnegative integers, and $f_1,f_2,\ldots,f_s$ are pairwise distinct Fermat primes. Only such values of $q$ offer a nontrivial number of $p$ satisfying the required property. (For most $q$, the only $p$ that works is $p=0$.) This is due to a theorem by Gauss (see here).

P.S.: By "most $q$," I lied. The natural density of $q$ with more than one working value of $p$ is somewhere (strictly) between $\frac{1}{2}$ and $\frac{3}{4}$.

Batominovski
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  • https://math.stackexchange.com/questions/2522482/also-is-there-a-programming-language-that-can-easily-tell-of-a-number-is-a-radi – User3910 Nov 16 '17 at 03:05