# Generate a polynomial w/ integer coefficients whose roots are rational values of sine/cosine?

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...

• Look up Chebyshev polynomials. This isn't appropriate for MO, but if you want more clarification you should ask on math.stackexchange.com. – 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 mathoverflow.net/questions/62080/… – Johan Wästlund Oct 23 '12 at 8:12

Let $$T_n$$ and $$U_n$$ be the Chebyshev 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.
• 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