If z1=cosθ+i sinθ and 1, z1,z12,z13,........,z1n−1 are vertices of a regular polygon such that Im(z12)Re(z1)=5−12, then the sum of the digits of n is

z1=cosθ+isinθ and 1,z1,z12,z13,.............z1n−1 are vertices of a regular polygon θ=2πn,z1=cos2πn+i sin 2πn,z12=ei4π4+i sin4πn lm(z12)Re(z1)=5−12 ⇒sin4πncos(2πn)=5−12 ⇒2sin(2πn)cos(2πn)cos(2πn)=5−12 ⇒sin(2πn)=5−14=sin180=sinπ10 ⇒ 2πn=π10 ⇒n=20

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If $z_{1} = \cos θ + i \sin θ a n d 1, z_{1}, z_{1}^{2}, z_{1}^{3}, ........, {z_{1}}^{n - 1}$ are vertices of a regular polygon such that $\frac{I_{m} (z_{1}^{2})}{R_{e} (z_{1})} = \frac{\sqrt{5} - 1}{2},$ then the sum of the digits of n is

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Detailed Solution

$z_{1} = \cos θ + i \sin θ a n d 1, z_{1}, z_{1}^{2}, z_{1}^{3}, ............. {z_{1}}^{n - 1}$ are vertices of a regular polygon
$θ = \frac{2 π}{n}, z_{1} = \cos \frac{2 π}{n} + i \sin \frac{2 π}{n}, z_{1}^{2} = e^{i 4 \frac{π}{4}} + i \sin \frac{4 π}{n} \frac{l m (z_{1}^{2})}{Re (z_{1})} = \frac{\sqrt{5} - 1}{2} \Rightarrow \frac{\sin \frac{4 π}{n}}{\cos (\frac{2 π}{n})} = \frac{\sqrt{5} - 1}{2} \Rightarrow \frac{2 \sin (\frac{2 π}{n}) \cos (\frac{2 π}{n})}{\cos (\frac{2 π}{n})} = \frac{\sqrt{5} - 1}{2} \Rightarrow \sin (\frac{2 π}{n}) = \frac{\sqrt{5} - 1}{4} = \sin 18^{0} = \sin \frac{π}{10} \Rightarrow \frac{2 π}{n} = \frac{π}{10} \Rightarrow n = 20$