Example 2. If the difference between the two roots of the equation $\mathrm{x}^{2}-5 \mathrm{x} \cdot \sin \theta+1=0$ is $2 \sqrt{3}$, and $180^{\circ}<\theta<270^{\circ}$, find the value of $\operatorname{ctg} \theta$.
$$
=\sqrt{2} \sin ^{2} \theta-4=2 \sqrt{3} \text {. }
$$

Given: $\sin ^{2} \theta=\frac{13}{25}$.
Also, $180^{\circ}<\theta<270^{\circ}$, so take $\sin \theta=-\frac{4}{5}$, at this time $\cos \theta=-\sqrt{1-\left(-\frac{4}{5}\right)^{2}}=-\frac{3}{5}$. Therefore, $\operatorname{ctg}=\frac{\cos \theta}{\sin \theta}=\frac{3}{4}$.