Let \(I_{n}=\int_{0}^{\frac{1}{2} \pi} x^{n} \sin x \mathrm{~d} x\).
(i) Prove that, for \(n \geqslant 2\),
\(I_{n}+n(n-1) I_{n-2}=n\left(\frac{1}{2} \pi\right)^{n-1}\)

(ii) Calculate the exact value of \(I_{1}\) and deduce the exact value of \(I_{3}\).