Here is a function that I have had fun with: if you type in solve minimalpolynomial[sin[pi/x]]=0 into Wolfram Alpha and type in x as a number, otherwise it will not know how to interpret your input, you can get the sum and product of the roots. Here one of the more subjectively interesting parts ...
The notation in your defining equation for $P(x)$ makes no sense. Also, note that most computer algebra systems work on the (very bad) garbage-in/garbage-out principal, so if you ask for the minimal polynomial of a transcendental number you won't get a useful result.
@Blue essentially, this means that the product of roots is equal to $(-1)^{\mathrm{degree}}\frac{\mathrm{constant\space term}}{\mathrm{highest\space degree \space term}}$ based on the wikipedia page?
@RobArthan, as you can see, the question does not prioritize finding roots of transcendental sine value polynomials nor does it talk about them in the latest version of this question. Please look at the graph of the function to see that the notation does indeed make sense in a way.
Your edits do not address any of my concerns: your notation in your proposed definition of $P(x)$ still makes no sense and you are still talking about the minimal polynomials of real numbers that may be transcendental.
@TymaGaidash It helped if you wrote down what $P(x)$ is for $x=3,4,5$ for example. It is unclear what you mean by "polynomial" and "product". And parts of the question make no sense e.g. it is not possible for a (non-constant) polynomial to be bounded or have a (finite) limit at $\pm\infty$.
@dxiv, the graph has the first few values of the function. If you believe that the function name is confusing, then please suggest a new name also, it was written explicitly that “$x\in\Bbb Q$“ or x is a real rational number to avoid hyperbolic functions from imaginary values and transcendental values. Do you have any other questions?
@TymaGaidash: Correct! If you do a web search, you'll find various articles about the coefficients of minimal polynomials of trig functions. (Expect to see terms like "cyclotomic polynomial" and "chebyshev polynomial".)
@TymaGaidash You need to explain in the question what those values mean and how you derived them. While doing that you should realize that $P(x)$ is not a polynomial, and calling it a polynomial only adds to the confusion about what you actually mean.
@TymaGaidash: I re-posted because I'd inadvertently suggested that you were interested in only the constant term (which is false; minimal polynomials are obviously not always monic).
@dxiv I thought I posted that these were gotten from Wolfram. I guess I simply forgot, as for P(x), I meant that it was named after the polynomials for which it was related to but not necessarily was a polynomial. I have changed it to the “generic” f(x).
@TymaGaidash The minimal polynomial of $\cos(\pi / n)$ is a factor of a Chebyshev polynomial of the first kind. Determining the first and last coefficients would take some work, however, and doing the same for $\sin(\pi / n)$ is somewhat more complicated still.
BTW, you're trying too hard to present a symbolic definition. Simplify: "For $q\in\Bbb{Q}$, define $f(q)$ to be the product of the roots of the minimal polynomial of $\sin(\pi/q)$." Done! This avoids unnecessary references to $n$, $N$, the non-conventional $[y]_n$, etc. You can go on to observe: "Writing $m$ for the minimal polynomial, we have $$f(q) = (-1)^{\deg(m)} \frac{\text{trailing coefficient of $m$}}{\text{leading coefficient of $m$}}$$ by Vieta's formulas." (Also, there's no "probably" about expressing $f(q)$ in terms of the coeffs and degree. Vieta's formulas are well-known.)
@Blue, thanks, I will include the other definition, which has wolfram mathworld’s root notation, for another perspective. It is all now up to finding the factor of Chebyshev polynomials as dxiv said. Maybe I will use cosine instead of sine for easier evaluation and then find the sum analog of this function or a vieta generalization.
Discussion on question by Tyma Gaidas…
Discussion on question by Tyma Gaidas…