![enter image description here][1]

Above you see a $12$-frame  animation  of the family of  trig polynomials

$$ P_k(t) =\cos t+\cos 2t+\cos 3t+\cos kt,\;\;\; k=4,\dotsc, 15. $$


I have included it to illustrate  the fact that  there seems to be another quantity relevant to your question, besides $n$, namely the degree $d=\max\{k_i;\;\;i=1.\dotsc, n\}$. In the above example $n=4$, but the degree varies from $4$ to $15$.

The next animation  may be more suggestive because you can  see large intervals where  the trig polynomials are negative. More precisely, below is a $10$-frame animation of the trig polynomials

$$P_k(t) =\cos 2t+\cos 3t+ \cos(4k+1)t, \;\;k=4,\dotsc, 13. $$


![enter image description here][2]


  [1]: https://i.sstatic.net/xPvnC.gif
  [2]: https://i.sstatic.net/pRoLb.gif