Complex Numbers (id: 8b13945fa)

本题目来源于试卷: Complex Numbers，类别为 IB数学

[问答题]
1. Use de Moivre's theorem to 1v 8qu6u7r4epw vs0 wzfind th oe5bf4-(yj-0wpyzl hky f-4q e value of ${[\cos \left(\frac{\pi }{6}\right)+\mathrm{i}\sin \left(\frac{\pi }{6}\right)]}^{12}$ .
2. Use mathematical induction to prove that

$(\cos \alpha -\mathrm{i}\sin \alpha {)}^n=\cos (n\alpha )-\mathrm{i}\sin (n\alpha )\text{ for all }n\in {\mathbb{Z}}^{+}\text{. }$

eet $w=\cos \alpha +\mathrm{i}\sin \alpha$ .
3. Find an expression in terms of $\alpha$ for $w^n-{\left(w^{\ast }\right)}^n$, $n\in {\mathbb{Z}}^{+}$ , where $w^{\ast }$ is the complex conjugate of w .
4. 1. Show that $w{w}^{\ast }=1$ .
2. Write down and simplify the binomial expansion of ${(w-w^{\ast })}^3$ in terms of w and $w^{\ast }$ .
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 $0\le \alpha \le \pi$ .

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