The Long Run Guess

[random 8]

Recap:

• The number is algebraic and irrational.
  
• The number is not real or imaginary (i.e. in rectangular form [a+bi, a∈ℝ, b∈ℝ], a and b are both non-zero).
  
• The number, written in polar form (re^iθ, r∈ℝ, 0≤θ<2π), contains π.
  
• r∈ℤ, r ≠ 5, r ≠ 10, |a| ≤ 100, |b| ≤ 100.
  
• tan(|θ|) ∉ A and cot(|θ|) ∉ A, where A = {3/4, 5/12, 7/24, 8/15, 9/40, 11/60, 12/35, 13/84, 16/63, 20/21, 20/99, 28/45, 33/56, 36/77, 39/80, 48/55, 60/91, 65/72}.
  

New hint: the number is a root of a non-zero
polynomial in one variable
with integer coefficients and
degree no greater than 5.