Question

If `z_1 and bar z_1` represent adjacent vertices of a regular polygon of n sides where centre is origin and if `(Im(z))/(Re(z)) = sqrt(2) - 1`, then n is equal to: (A) 8 (B) 16 (C) 24 (D) 32

Answer 1

Please refer to the diagram in the video.
From the diagram,
`tan theta = y/x`
`tan theta = sqrt2-1`
Now, `(2pi)/n = 2theta`
`=>tan((2pi)/n) = tan2theta`
`=>tan((2pi)/n) = (2tantheta)/(1-tan^2theta)`
`=>tan((2pi)/n) = (2(sqrt2-1))/(1-(sqrt2-1)^2)`
`=>tan((2pi)/n) = (2(sqrt2-1))/(1-(2+1-2sqrt2))`
`=>tan((2pi)/n) = (2(sqrt2-1))/(2(sqrt2-1))`
`=>tan((2pi)/n) = 1 = tan(pi/4)`
`=>(2pi)/n = pi/4`
`=> n = 8`
So, option `A` is the correct option.