(i) The graph of \(y = \sin x\) starts at \((0,0)\), peaks at \((90^\circ, 1)\), and returns to \((180^\circ, 0)\). The graph of \(y = \cos 2x\) starts at \((0, 1)\), completes one full cycle by \(180^\circ\), and oscillates between -1 and 1.

(ii) To verify \(x = 30^\circ\) is a root, calculate \(\sin 30^\circ = 0.5\) and \(\cos 60^\circ = 0.5\), confirming \(\sin 30^\circ = \cos 60^\circ\). The other root is found by solving \(\sin x = \cos 2x\) for \(0^\circ \leq x \leq 180^\circ\), giving \(x = 150^\circ\).

(iii) The inequality \(\sin x < \cos 2x\) holds for \(0 \leq x < 30\) and \(150 < x \leq 180\).