The inequality $$3 + 4 \cos \theta + \cos 2 \theta \geq 0$$ plays a key role in the proof of the classical zero-free region of the Riemann zeta function. Are there other inequalities of the form $$\sum_{i=0}^k a_i \cos b_i \theta \geq 0,\;\;\;\;\;a_\geq 0$$ such that $a_{i_0} = \sum_{i\ne i_0} a_i$ for some $0\leq i_0\leq k$ and $a_{0} < \frac{3}{4} a_{i_0}$?