(i) Expand \(2 \sin(x - 30^\circ)\) using the angle subtraction formula:

\(2 \sin(x - 30^\circ) = 2(\sin x \cos 30^\circ - \cos x \sin 30^\circ)\)

\(= \sqrt{3} \sin x - \cos x\)

Combine terms: \(\sqrt{3} \sin x - 2 \cos x\).

Using the identity:

\(a \sin \theta + b \cos \theta \equiv R \sin(\theta + \alpha)\)

\(R = \sqrt{a^2 + b^2} = \sqrt{(\sqrt{3})^2 + (-2)^2} = \sqrt{7}\)

\(\tan \alpha = \frac{b}{a} = \frac{-2}{\sqrt{3}}\)

\(\alpha = \tan^{-1}\left(\frac{-2}{\sqrt{3}}\right) \approx 49.11^\circ\)

(ii) Solve \(\sqrt{3} \sin x - 2 \cos x = 1\):

\(R \sin(x - \alpha) = 1\)

\(\sqrt{7} \sin(x - 49.11^\circ) = 1\)

\(\sin(x - 49.11^\circ) = \frac{1}{\sqrt{7}}\)

\(x - 49.11^\circ = \sin^{-1}\left(\frac{1}{\sqrt{7}}\right) \approx 22.21^\circ\)

\(x = 22.21^\circ + 49.11^\circ = 71.3^\circ\)