The first value of \(\theta\) satisfying the equations \[ \tan \theta = \frac{1}{\sqrt{3}}, \quad \sin \theta = -\frac{1}{2} \] lies in the third quadrant, and the value is \[ \pi + \frac{\pi}{6} = \frac{7\pi}{6} \] The general value of \(\theta\) can be obtained by adding a multiple of \(2\pi\): \[ \text{General value} = 2n\pi + \frac{7\pi}{6} \] \