Part (i):

Given the equation:

\(\sin(x - 60^\circ) = 3 \cos(x - 45^\circ)\)

Using the angle subtraction identities:

\(\sin(x - 60^\circ) = \sin x \cos 60^\circ - \cos x \sin 60^\circ\)

\(\cos(x - 45^\circ) = \cos x \cos 45^\circ + \sin x \sin 45^\circ\)

Substitute the exact values:

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

Rearrange and simplify:

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

Combine terms:

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

\(\sin x \left(1 - 3\sqrt{2}\right) = \cos x \left(3\sqrt{2} + \sqrt{3}\right)\)

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

\(\tan x = \frac{-(6+\sqrt{6})}{(6-\sqrt{2})}\)

Part (ii):

Using the result from part (i), solve for \(x\):

\(x = 118.5^\circ, 298.5^\circ\)