(i) Start with the given equation:

\(\sin(\theta + 45^\circ) + 2 \cos(\theta + 60^\circ) = 3 \cos \theta\)

Use angle addition formulas:

\(\sin(\theta + 45^\circ) = \sin \theta \cos 45^\circ + \cos \theta \sin 45^\circ\)

\(\cos(\theta + 60^\circ) = \cos \theta \cos 60^\circ - \sin \theta \sin 60^\circ\)

Substitute \(\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 60^\circ = \frac{1}{2}\), and \(\sin 60^\circ = \frac{\sqrt{3}}{2}\):

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

Simplify:

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

Combine terms:

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

Rearrange:

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

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

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

\(\tan \theta = \frac{2\sqrt{2} - 1}{1 - \sqrt{6}}\)

(ii) Solve for \(\theta\):

Using the expression for \(\tan \theta\), find \(\theta\) in the range \(0^\circ < \theta < 360^\circ\).

\(\theta = 128.4^\circ\) and \(\theta = 308.4^\circ\)