(i) Let \(z=\cos \theta+\mathrm{i} \sin \theta\). Show that \(z-\frac{1}{z}=2 \mathrm{i} \sin \theta\) and hence express \(16 \sin ^{5} \theta\) in the form \(\sin 5 \theta+p \sin 3 \theta+q \sin \theta\), where \(p\) and \(q\) are integers to be determined.

(ii) Hence find the exact value of \(\int_{0}^{\frac{1}{3} \pi} 16 \sin ^{5} \theta \mathrm{~d} \theta\).