本题目来源于试卷: Proofs Proofs ，类别为 IB数学
[问答题]
1. Use de Moivre's theorem twn d w) gp2g.a16lu9qrbrpa;),tibxb8.1jp oo find the value of $ \left[\cos \left(\frac{\pi}{6}\right)+\mathrm{i} \sin \left(\frac{\pi}{6}\right)\right]^{12}$ .
2. Use mathematical induction to prove that
$(\cos \alpha-\mathrm{i} \sin \alpha)^{n}=\cos (n \alpha)-\mathrm{i} \sin (n \alpha) \quad $for all $ n \in \mathbb{Z}^{+} $.
Let $w=\cos \alpha+\mathrm{i} \sin \alpha$ .
3. Find an expression in terms of $\alpha$ for $ w^{n}-\left(w^{*}\right)^{n}$, $n \in \mathbb{Z}^{+}$ , where $ w^{*}$ is the complex conjugate of w .
4. 1. Show that $w w^{*}=1$ .
2. Write down and simplify the binomial expansion of $ \left(w-w^{*}\right)^{3}$ in terms of w and $w^{*}$ .
3. Hence show that $ \sin (3 \alpha)=3 \sin \alpha-4 \sin ^{3} \alpha$ .
5. Hence solve $4 \sin ^{3} \alpha+(2 \cos \alpha-3) \sin \alpha=0$ for $0 \leq \alpha \leq \pi$ .
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