Functional Complex Trignometry

Geometry Level 4

The polynomial f(x)=x52+a1x51+a2x50++a51x+a52 f(x) = x^{52} + a_1x^{51} + a_2x^{50} + \cdots + a_{51}x+a_{52} has roots tanθ1,tanθ2,tanθ3,,tanθ52.\tan\theta_1, \tan\theta_2, \tan\theta_3, \ldots, \tan\theta_{52}.

We are also told that:

  • f(i)=i(2+3)+1f(-i) = i(2 + \sqrt3) + 1 ,
  • f(1)=i3+1 f(-1) = i\sqrt3 + 1 ,
  • f(i)=i(2+3)1 f(i) = i(2 + \sqrt3) - 1 ,
  • f(1)=i31 f(1) = i\sqrt3 - 1, and
  • f(0)=1 f(0) = 1.

Find the minimum positive value of θ1+θ2+θ3++θ52.\theta_1 + \theta_2 +\theta_3 + \cdots +\theta_{52}.


All of my problems are original.
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