(i) Start with the equation \(\sin 2x + 3 \cos 2x = 0\).

Rearrange to \(\tan 2x = -3\).

Find \(2x\) using the inverse tangent: \(2x = 180^\circ - 71.6^\circ\) or \(360^\circ - 71.6^\circ\).

This gives \(2x = 108.4^\circ\) or \(2x = 288.4^\circ\).

Divide by 2 to find \(x\): \(x = 54.2^\circ\) or \(x = 144.2^\circ\).

Also consider the periodicity of tangent: \(x = 234.2^\circ\) and \(x = 324.2^\circ\).

(ii) The range \(0^\circ \leq x \leq 1080^\circ\) is three times the range \(0^\circ \leq x \leq 360^\circ\), so there are 3 times the number of solutions: 12 answers.