(i) Separate variables: \(\frac{1}{x} dx = \frac{\sin 2\theta}{3 + \cos 2\theta} d\theta\).

Integrate both sides: \(\int \frac{1}{x} dx = \int \frac{\sin 2\theta}{3 + \cos 2\theta} d\theta\).

This gives \(\ln x = -\frac{1}{2} \ln(3 + \cos 2\theta) + C\).

Use initial condition \(x = 3\) when \(\theta = \frac{1}{4}\pi\) to find \(C\):

\(\ln 3 = -\frac{1}{2} \ln(3 + \cos \frac{\pi}{2}) + C\)

\(\ln 3 = -\frac{1}{2} \ln 3 + C\)

\(C = \frac{3}{2} \ln 3\)

Thus, \(\ln x = -\frac{1}{2} \ln(3 + \cos 2\theta) + \frac{3}{2} \ln 3\)

Rearrange to find \(x\):

\(x = \sqrt{\frac{27}{3 + \cos 2\theta}}\)

(ii) The least value of \(x\) occurs when \(\cos 2\theta = 1\), giving:

\(x = \sqrt{\frac{27}{3 + 1}} = \frac{3\sqrt{3}}{2}\)