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I'm a high school calculus/precalculus teacher, so forgive me if the question is a little basic. One of my (very gifted) students recently came up with a construction yielding a quartic, one of whose roots was sin(80º) -- which led me to the startling discovery that this (and, indeed, all rational values of sine/cosine (in degrees; that is, rational multiples of π)) are algebraic.

I've come across a number of proofs that the numbers are algebraic since, which, as I understand it, goes back to complex roots of unity. What I -haven't- seen, and would very much like to see/understand, is some general method for generating/constructing polynomials (w/ integer coefficients) whose roots are sine/cosine of rational values (in degrees). (My student's method only works for 80º/10º, 70º/20º, and 75º/15º, unfortunately). Would much appreciate...

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Look up Chebyshev polynomials. This isn't appropriate for MO, but if you want more clarification you should ask on –  Qiaochu Yuan Oct 22 '12 at 1:35
Another keyword would be "cyclotomic polynomials", although these are not quite exactly what you want. –  paul garrett Oct 22 '12 at 1:36
Qiaochu, the answer is well-known and easy to google if you know the right keywords. Still, I think the question is far from inappropriate. It's clearly stated, with a description of how it came up and the OP's efforts to solve it. Yes it's homework, but give the poor teacher a break... –  Johan Wästlund Oct 23 '12 at 8:08
By the way, a related question is… –  Johan Wästlund Oct 23 '12 at 8:12

1 Answer 1

Let $T_n$ and $U_n$ be the Chebychev polynomials of the first and second kind, respectively.

Let $\psi_n(x)$ be the minimal polynomial of the algebraic integer $2 \cos \frac{2 \pi}{n}$. Then

$$ U_n(x)=\prod_{\substack{ j|2n+2 \\ j\not=1,2}} \psi_j(2x) \ . $$ Let $n=2^{\alpha} N$ where $N$ is odd and let $r=2^{\alpha+2}$. Then

$$ T_n(x)=\frac{1}{2}\prod_{\substack{ j|N \\ }} \psi_{r j}(2x) \ . $$

The irreducible polynomials $\psi_n$ were introduced by Lehmer in D. H. Lehmer A Note on Trigonometric Algebraic Numbers. Amer. Math. Monthly,40 (1933) 165-166.

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For historical accuracy, we should note that Lagrange and Vandermonde apparently knew quite a lot about cyclotomic polynomials c. 1770, from which it is not difficult to obtain polynomials with such values of trig functions as zeros, and throughout the 19th century many people studied these things intensely. Hopefully that "Monthly" article is readable, and has a sensible bibliography giving historical indicators. –  paul garrett Oct 22 '12 at 19:59
Yes! In his proof he uses cyclotomic polynomials. A cyclotomic polynomial can be written as $\Phi_n(x)=x^d \psi(x+x^{-1})$ where $d$ is the degree of $\psi$ and then it follows that $\psi$ is irreducible. In the Monthly note he quotes Sylvester and Kronecker. –  Pantelis Damianou Oct 22 '12 at 20:11

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