回復 28# kittyyaya 的帖子
填充第 15 題：
設 \(O,P,Q,R\) 分別表示在複數平面上的原點、\(z_1, z_2, z_1+z_2\)，
設 \(\theta=\angle{POQ}\)
則 \(\overline{OR}^2 = \overline{OP}^2+\overline{PR}^2-2\cdot\overline{OP}\cdot\overline{PR}\cdot\cos\left(\pi-\theta\right)\)
\(\Rightarrow 7 = 3^2+5^2-2\cdot3\cdot5\cdot\cos\left(\pi-\theta\right)\)
\(\displaystyle\Rightarrow \cos\theta=\frac{1}{2}\)
\(\displaystyle\Rightarrow \theta=\frac{\pi}{3}\)
因此，
\(\displaystyle\left(\frac{z_2}{z_1}\right)^3=\left(\frac{5}{3}\left(\cos\left(\pm\frac{\pi}{3}\right)+i\sin\left(\pm\frac{\pi}{3}\right)\right)\right)^3\)
\(\displaystyle=\frac{125}{27}\left(\cos\left(\pm\pi\right)+i\sin\left(\pm\pi\right)\right)\)
\(\displaystyle=-\frac{125}{27}\)
