To solve the equation \(\sin(\theta - 30^\circ) + \cos \theta = 2 \sin \theta\), we start by using the angle subtraction identity:

\(\sin(\theta - 30^\circ) = \sin \theta \cos 30^\circ - \cos \theta \sin 30^\circ\)

Substitute the values \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\sin 30^\circ = \frac{1}{2}\):

\(\sin \theta \cdot \frac{\sqrt{3}}{2} - \cos \theta \cdot \frac{1}{2} + \cos \theta = 2 \sin \theta\)

Simplify the equation:

\(\frac{\sqrt{3}}{2} \sin \theta - \frac{1}{2} \cos \theta + \cos \theta = 2 \sin \theta\)

Combine like terms:

\(\frac{\sqrt{3}}{2} \sin \theta + \frac{1}{2} \cos \theta = 2 \sin \theta\)

Rearrange to form:

\(\frac{\sqrt{3}}{2} \sin \theta - 2 \sin \theta = -\frac{1}{2} \cos \theta\)

Factor out \(\sin \theta\):

\(\sin \theta \left( \frac{\sqrt{3}}{2} - 2 \right) = -\frac{1}{2} \cos \theta\)

Divide both sides by \(\cos \theta\):

\(\tan \theta = \frac{1}{4 - \sqrt{3}}\)

Calculate \(\theta\):

\(\theta = \tan^{-1}\left(\frac{1}{4 - \sqrt{3}}\right) \approx 23.8^\circ\)

Thus, the solution is \(\theta = 23.8^\circ\).