Subject: Mathematics — Complex Numbers
Question: Let $w = \frac{\sqrt{3} + i}{2}$ and
$P = \left\{ w^{n} \cdot n = 1,2,3,\ldots \right\}$. Further
$H_{1} = \left\{ z \in \mathbb{C:}Rez > \frac{1}{2} \right\}$ and
$H_{2} = \left\{ z \in \mathbb{C \cdot}Rez < \frac{- 1}{2} \right\}$,
where c is the set of all complex numbers. If
$z_{1} \in P \cap H_{1},z_{2} \in P \cap H_{2}$ and $O$ represents the
origin, $\angle z_{1}Oz_{2} =$ when $\alpha = (\frac{\pi}{2},\ \pi)$
will be
Answer: $\frac{2\pi}{3}\ ,\ \frac{5\pi}{6}$
Solution: We have
$w = \frac{\sqrt{3} + i}{2} = cos\frac{\pi}{6} + isin\frac{\pi}{6}$\
$$\Rightarrow w^{n} = cos\frac{n\pi}{6} + isin\frac{n\pi}{6}
$$$\therefore P$ contains all those points which lie on unit circle and
have arguments $\frac{\pi}{6},\frac{2\pi}{6},\frac{3\pi}{6}$ and so on.
[IMAGE] Since, $z_{1} \in P \cap H_{1}$ and $z_{2} \in P \cap H_{2}$,
therefore $z_{1}$ and $z_{2}$ have possible positions as shown in the
figure,
$\therefore\angle z_{1}{Oz}_{2}$ can be $\frac{2\pi}{3}$ or
$\frac{5\pi}{6}$.