4. Let $0 \leqslant \theta \leqslant \frac{\pi}{2}$, the range of real numbers $m$ that satisfy the inequality $\sin ^{2} \theta+3 m$ - $\cos \theta-6 m-4<0$ is $\qquad$ .
4. $m>-\frac{1}{2}$.

The inequality becomes $3 m>\frac{3+\cos ^{2} \theta}{\cos \theta-2}$ for all $0 \leqslant \theta \leqslant \frac{\pi}{2}$, so $3 m$ should be greater than the maximum value of $f(\theta)=-\frac{3+\cos ^{2} \theta}{2-\cos \theta}$. It is easy to see that $-\frac{3+\cos ^{2} \theta}{2-\cos \theta} \leqslant-\frac{3}{2}$, hence $3 m>-\frac{3}{2}$, which means $m>-\frac{1}{2}$.