本题目来源于试卷: Complex Numbers，类别为 IB数学
[问答题]
1. Express $-4+4 \sqrt{3} \mathrm{i}$ in the form r $e^{\mathrm{i} \theta}$ , where r>0 and $-\pi<\theta \leq \pi$ .
Let the roots of the equation $z^{3}=-4+4 \sqrt{3} \mathrm{i}$ be $z_{1}$, $z_{2}$ and $z_{3}$ .
2. Find $z_{1}$, $z_{2}$ and $ z_{3}$ expressing your answers in the form $ r e^{\mathrm{i} \theta}$ , where r>0 and $-\pi<\theta \leq \pi$ .
On an Argand diagram, $z_{1}$, $z_{2}$ and $z_{3}$ are represented by the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ , respectively.
3. Find the area of the triangle $\mathrm{ABC}$ .
4. By considering the sum of the roots $z_{1}, z_{2}$ and $z_{3}$ , show that
$\cos \left(\frac{2 \pi}{9}\right)+\cos \left(\frac{4 \pi}{9}\right)+\cos \left(\frac{8 \pi}{9}\right)=0$
参考答案: 
本题详细解析:
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