Prove by mathematical induction that, for every positive integer \(n\),
\((\cos \theta+\mathrm{i} \sin \theta)^{n}=\cos n \theta+\mathrm{i} \sin n \theta .\)

Express \(\sin ^{5} \theta\) in the form \(p \sin 5 \theta+q \sin 3 \theta+r \sin \theta\), where \(p, q\) and \(r\) are rational numbers to be determined.