Let w = \frac{\sqrt{3} + i}{2} and P = {wn : n= 1, 2, 3, …….}. Further H_{1} : \left{\right. z \in C : Re \textrm{ } z > \frac{1}{2} \left.\right} and H_{2} : \left{\right. z \in C : Re \textrm{ } z < \frac{- 1}{2} \left.\right}, where C is the set of all complex numbers. If z1 ∈ P ⋂ H1, z2 ∈ ⋂ H2 and O represents the origin, then ∠z1 O z2 =

Engineering

Mathematics

nth Roots of Unity

Question

Let w = $\frac{\sqrt{3} + i}{2}$ and P = {wⁿ : n= 1, 2, 3, …….}. Further $H_{1} : {z \in C : Re z > \frac{1}{2}}$ and $H_{2} : {z \in C : Re z < \frac{- 1}{2}}$ , where C is the set of all complex numbers. If z₁ ∈ P ⋂ H₁, z₂ ∈ ⋂ H₂ and O represents the origin, then ∠z₁O z₂ =

$\frac{5 π}{6}$

$\frac{π}{2}$

$\frac{2 π}{3}$

$\frac{π}{6}$

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$w = \frac{\sqrt{3} + i}{2} = \cos \frac{π}{6} + i \sin \frac{π}{6}$

$P = w^{n} = \cos \frac{nπ}{6} + isin \frac{nπ}{6}, n = 0, 1, 2, . ......., 11 .$

$H_{1} : {z \in C : Re (z) > \frac{1}{2}}$

$z_{1} \in \cap H_{1}$

$∴ possible value of z_{1} are {(\frac{\sqrt{3}}{2}, \frac{1}{2}), (0, 1), (\frac{\sqrt{3}}{2}, \frac{- 1}{2})}$

$H_{1} = {z \in C : Re (z) < \frac{- 1}{2}}$

$z_{2} \in p \cap H_{2}$

$∴ possible values of z_{2} are {(\frac{- \sqrt{3}}{2}, \frac{1}{2}), (- 1, 0), (\frac{- \sqrt{3}}{2}, \frac{- 1}{2})}$

$Now, possible angle z_{1} {Oz}_{2} is \frac{2 π}{3}, \frac{5 π}{6}, π$