While it is an old and fundamental fact that every (nice enough) even function 
$f : [-\pi,\pi] \rightarrow \mathbb{C}$ may be uniquely expressed as a cosine series 
\[  f(\theta) = \sum_{r \geq 0 } C_r\cos(r\theta), 
\] the relationship between the sequence of coefficients $(C_r)_{r \geq 0 }$ and the behavior of the function $f$
remains mysterious in many aspects. We mention two variations on this theme. First a more probabilistic setting: what can be said about a random variable if we constrain the roots of the probability generating function? We then settle on our main topic; a solution to a problem of J.E. Littlewood about 
the behavior of the zeros of cosine polynomials with coefficients $C_r \in \{0,1\}$.