Complex Numbers (id: 2845b5edd)

本题目来源于试卷: Complex Numbers，类别为 IB数学

[问答题]
1. Find the roots of z^nq-c6iu-8h o di*/eol{16}=udn3u :,kl k q(9u:jshb3 j8dd1 which satisfy the condition $0<\arg (z)<\frac{\pi }{2}$ , expressing your answer in the form $r{e}^{\mathrm{i}\theta }$ , where r, $\theta \in {\mathbb{R}}^{+}$ .
2. Let S be the sum of the roots found in part (a).
1. Show that $\mathrm{Re}(S)=\mathrm{Im}(S)$ .
2. By writing $\frac{\pi }{8}$ as $\frac{1}{2}\cdot \frac{\pi }{4}$ , find the value of $\cos \left(\frac{\pi }{8}\right)$ in the form $\frac{\sqrt{a+\sqrt{b}}}{c}$, where a, b and c are integers to be determined.
3. Hence, or otherwise, show that $S=\frac{1}{2}(\sqrt{2+\sqrt{2}}+\sqrt{2}+\sqrt{2-\sqrt{2}})(1+\mathrm{i})$ .

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