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If z1=cosx+isinx and 1,z1,z1^2,z1^3,.....…....,z1^n-1 are the vertices of a regular polygon such that I'm(z1^2)/Re(z1)=√5-1/2,then the value of n is

If z1=cosx+isinx and 1,z1,z1^2,z1^3,.....…....,z1^n-1 are the vertices of a regular polygon such that    I'm(z1^2)/Re(z1)=√5-1/2,then the value of n is

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1 Answers

Aditya Gupta
2081 Points
4 years ago
note that z1= e^ix
now, 1, e^ix, e^i2x, …......, e^i(n – 1)x are the vertices of a regular polygon.
so distance b/w 1 and e^ix= distance b/w 1 and e^i(n – 1)x
or |e^ix – 1|= |e^i(n – 1)x – 1|
let (n – 1)x= y
so |cosx – 1+isinx|= |cosy – 1+isiny|
(cosx – 1)^2+sin^2x= (cosy – 1)^2+sin^2y
or cosx= cosy
or x+y= 2pi
or x+(n – 1)x= 2pi
or n= 2pi/x
now, given  I'm(z1^2)/Re(z1)=√5-1/2
or  I'm(e^i2x)/cosx=√5-1/2
or sin2x/cosx= √5-1/2
or 2sinxcosx/cosx= √5-1/2
or sinx= √5-1/4= sin 18 deg
so x= 18deg= pi/10 rad
so n= 2pi/(pi/10)
or n= 20.
kindly approve :)

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