_{this is an answer to the question without the additional condition that $b_0=0$}

for example, $$6+\cos \theta+2 \cos 2 \theta+3 \cos 3 \theta\geq 0,$$ or more generally $$\tfrac{1}{2}k(k+1)+\sum_{n=1}^k n \cos n\theta\geq 0.$$

_{this is an answer to the question as originally posed, without the additional conditions on $a_0$ and $b_0$}

for example, $$6+\cos \theta+2 \cos 2 \theta+3 \cos 3 \theta\geq 0,$$ or more generally $$\tfrac{1}{2}k(k+1)+\sum_{n=1}^k n \cos n\theta\geq 0.$$

for example, $$6+\cos \theta+2 \cos 2 \theta+3 \cos 3 \theta\geq 0,$$ or more generally $$\tfrac{1}{2}k(k+1)+\sum_{n=1}^k n \cos n\theta\geq 0.$$

_{this is an answer to the question without the additional condition that $b_0=0$}

for example, $$6+\cos \theta+2 \cos 2 \theta+3 \cos 3 \theta\geq 0,$$ or more generally $$\tfrac{1}{2}k(k+1)+\sum_{n=1}^k n \cos n\theta\geq 0.$$