cos_pi_div_three — Mathlib

English

cos(π/32) = √(2 + √(2 + √(2 + √2))) / 2.

Русский

cos(π/32) = √(2 + √(2 + √(2 + √2))) / 2.

LaTeX

$$$\\cos\\left(\\frac{\\pi}{32}\\right) = \\frac{\\sqrt{2 + \\sqrt{2 + \\sqrt{2 + \\sqrt{2}}}}}{2}$$$

Lean4

/-- The cosine of `π / 3` is `1 / 2`. -/
@[simp]
theorem cos_pi_div_three : cos (π / 3) = 1 / 2 :=
  by
  have h₁ : (2 * cos (π / 3) - 1) ^ 2 * (2 * cos (π / 3) + 2) = 0 :=
    by
    have : cos (3 * (π / 3)) = cos π := by
      congr 1
      ring
    linarith [cos_pi, cos_three_mul (π / 3)]
  rcases mul_eq_zero.mp h₁ with h | h
  · linarith [pow_eq_zero h]
  · have : cos π < cos (π / 3) := by refine cos_lt_cos_of_nonneg_of_le_pi ?_ le_rfl ?_ <;> linarith [pi_pos]
    linarith [cos_pi]

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