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Let $p$ be a prime and $\theta \in \left[0, \frac{\pi}{2}\right]$ be a real number. Suppose $\cos \theta$ and $\sin \theta$ are algebraic over $\mathbb{Q}$. When do they also exist over $\mathbb{F}_p$, i.e. when do their $\mathbb{Q}$-minimal polynomials have a root mod $p$?

For instance, if $\theta = \frac{\pi}{3}$, then these polynomials are $2x-1$ and $4x^2 - 3$ and both simultaneously have a root if and only if $p\geq 3$ and $3$ is a quadratic residue mod $p$, i.e. if and only if $p\equiv \pm 1 \pmod {12}$.

Can this be nicely generalized?

DesmondMiles
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    Can you clarify your intention, please? Every polynomial with integer coefficients can be thought to have coefficients in the prime field $\Bbb{F}_p$ as well (some complications when the leading coefficient is divisible by $p$, but ignore those for now). In this sense "numbers" remain algebraic after reduction modulo $p$, but do require reinterpretation. Anyway, in the question body you seem to want the polynomials to have a root in the prime field. Which would be analogous to either $\cos\theta$ or $\sin\theta$ to be rational rather than just algebraic! – Jyrki Lahtonen Dec 12 '23 at 04:36
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    The obvious thing that can be said immediately is that $\Bbb{F}_p$ has roots of unity of order $n$ if and only if $n\mid p-1$. When that is the case we automatically have the counterparts of the trig functions when $\theta=2\pi k/n$, for all $k$. Because the trig functions at such a choice of $\theta$ are linear combinations of roots of unity, we have their counterparts in $\Bbb{F}_p$ more often than that. This already shows in your inclusion of the primes $p\equiv -1\pmod{12}$. – Jyrki Lahtonen Dec 12 '23 at 04:42
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    (cont'd) And we can say that, in this sense, $2\cos(2\pi k/n)$ exists in $\Bbb{F}_p$ whenever $n\mid (p^2-1)$. Something similar can be said about the sines, but a bif of extra care required to handle the fourth roots of unity. – Jyrki Lahtonen Dec 12 '23 at 04:44
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    (cont'd) But the arguments involving roots of unity miss out on other algebraic numbers on the unit circle of the complex plane. Those may become a bit hairy. At least I need more coffee before I begin to think about them :-) – Jyrki Lahtonen Dec 12 '23 at 04:45
  • Thanks, edited, my question was more about when do the trig functions exist mod $p$, thanks for your nice class of examples. – DesmondMiles Dec 12 '23 at 06:23
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    I think this is a rather academic question. The trigonometric functions are what it says: 3 angles plus a metric. It doesn't make much sense since finite fields don't have an interesting metric. The functions with the conditions you have to put in place make them a kind of a black box. It would be more honest to work with the power series in the first place and ask for their images in $\mathbb{F}_p[\theta].$ – Marius S.L. Dec 12 '23 at 06:53
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    You may want to also look at this or this. May be also this? In other words, on-site search FTW. – Jyrki Lahtonen Dec 12 '23 at 10:50
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    @MariusS.L. I think that using minimal polynomials is more generalizable. True, in a finite field we have no way of distinguishing, say, one primitive fifth root of unity from another. Consequently, no way of saying which element of a suitable field should play the role of $\cos(2\pi/5)$. The polynomial relations between $\cos(2\pi k/5)$, $k=1,2,3,4$ can be used to describe similar relations between those elements of a finite field, but no meaningful way of identifying those "cosines". – Jyrki Lahtonen Dec 12 '23 at 10:56
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    (cont'd) But, I don't see a way to use the (formal) power series of trig functions. Sooner rather than later those factorials become divisible by $p$, and we will be attempting to divide by zero. – Jyrki Lahtonen Dec 12 '23 at 10:56

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