A polynomial is called reciprocal if its coefficients read forwards are the same as those read backward - there is an obvious translation of reciprocal polynomials into cosine polynomials (since a term like $a_i x^{n + k}+ a_i x^{n-k}$ can be written like $a_i \exp(i n t) \cos 2\pi i k t,$ under the obvious substitution. It is known that a random reciprocal polynomial with i.i.d. Gaussian coefficients has 58% (= $1/\sqrt{3}$) of its roots on the unit circle, and experiments with i.i.d. uniform integer coefficients between -1000 and 1000, and degree 600 show that the smallest number of unimodular roots in 1000 trials is 290 (which is a lot).
It has been shown (Borwein, Peter; Erdélyi, Tamás Lower bounds for the number of zeros of cosine polynomials in the period: a problem of Littlewood. Acta Arith. 128 (2007), no. 4, 377–384.) that a family of reciprocal polynomials with bounded coefficients and growing degree will have more and more unimodular roots. So, the conclusion is that a reciprocal polynomial with no unimodular roots and of high degree will have large coefficients. But how large? And what are natural ways of producing these non-unimodular guys? (this question is strongly connected to Littlewood's conjectures, thus the harmonic analysis tag).
PS: Of course, only irreducible polynomials are really of interest. And the coefficients are integers, which, sadly, has remained unsaid above.
