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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

```
one year ago

```							note that z1= e^ixnow, 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)xor |e^ix – 1|= |e^i(n – 1)x – 1|let (n – 1)x= yso |cosx – 1+isinx|= |cosy – 1+isiny|(cosx – 1)^2+sin^2x= (cosy – 1)^2+sin^2yor cosx= cosyor x+y= 2pior x+(n – 1)x= 2pior n= 2pi/xnow, given  I'm(z1^2)/Re(z1)=√5-1/2or  I'm(e^i2x)/cosx=√5-1/2or sin2x/cosx= √5-1/2or 2sinxcosx/cosx= √5-1/2or sinx= √5-1/4= sin 18 degso x= 18deg= pi/10 radso n= 2pi/(pi/10)or n= 20.kindly approve :)
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one year ago
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