Documentation

Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

The arctan function. #

Inequalities, identities and Real.tan as a PartialHomeomorph between (-(π / 2), π / 2) and the whole line.

The result of arctan x + arctan y is given by arctan_add, arctan_add_eq_add_pi or arctan_add_eq_sub_pi depending on whether x * y < 1 and 0 < x. As an application of arctan_add we give four Machin-like formulas (linear combinations of arctangents equal to π / 4 = arctan 1), including John Machin's original one at four_mul_arctan_inv_5_sub_arctan_inv_239.

theorem Real.tan_add {x : } {y : } (h : ((∀ (k : ), x (2 * k + 1) * Real.pi / 2) ∀ (l : ), y (2 * l + 1) * Real.pi / 2) (∃ (k : ), x = (2 * k + 1) * Real.pi / 2) ∃ (l : ), y = (2 * l + 1) * Real.pi / 2) :
theorem Real.tan_add' {x : } {y : } (h : (∀ (k : ), x (2 * k + 1) * Real.pi / 2) ∀ (l : ), y (2 * l + 1) * Real.pi / 2) :
theorem Real.tan_two_mul {x : } :
Real.tan (2 * x) = 2 * Real.tan x / (1 - Real.tan x ^ 2)
theorem Real.continuous_tan :
Continuous fun (x : {x : | Real.cos x 0}) => Real.tan x

Real.tan as an OrderIso between (-(π / 2), π / 2) and .

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    noncomputable def Real.arctan (x : ) :

    Inverse of the tan function, returns values in the range -π / 2 < arctan x and arctan x < π / 2

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      @[simp]
      theorem Real.arctan_tan {x : } (hx₁ : -(Real.pi / 2) < x) (hx₂ : x < Real.pi / 2) :
      theorem Real.cos_sq_arctan (x : ) :
      Real.cos (Real.arctan x) ^ 2 = 1 / (1 + x ^ 2)
      theorem Real.arcsin_eq_arctan {x : } (h : x Set.Ioo (-1) 1) :
      @[simp]
      @[simp]
      theorem Real.arctan_eq_zero_iff {x : } :
      Real.arctan x = 0 x = 0
      theorem Real.arctan_eq_of_tan_eq {x : } {y : } (h : Real.tan x = y) (hx : x Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)) :
      @[simp]
      theorem Real.arccos_eq_arctan {x : } (h : 0 < x) :
      theorem Real.arctan_ne_mul_pi_div_two {x : } (k : ) :
      Real.arctan x (2 * k + 1) * Real.pi / 2
      theorem Real.arctan_add {x : } {y : } (h : x * y < 1) :
      Real.arctan x + Real.arctan y = Real.arctan ((x + y) / (1 - x * y))
      theorem Real.arctan_add_eq_add_pi {x : } {y : } (h : 1 < x * y) (hx : 0 < x) :
      theorem Real.arctan_add_eq_sub_pi {x : } {y : } (h : 1 < x * y) (hx : x < 0) :
      theorem Real.two_mul_arctan {x : } (h₁ : -1 < x) (h₂ : x < 1) :
      2 * Real.arctan x = Real.arctan (2 * x / (1 - x ^ 2))
      theorem Real.two_mul_arctan_add_pi {x : } (h : 1 < x) :
      2 * Real.arctan x = Real.arctan (2 * x / (1 - x ^ 2)) + Real.pi
      theorem Real.two_mul_arctan_sub_pi {x : } (h : x < -1) :
      2 * Real.arctan x = Real.arctan (2 * x / (1 - x ^ 2)) - Real.pi

      John Machin's 1706 formula, which he used to compute π to 100 decimal places.

      Real.tan as a PartialHomeomorph between (-(π / 2), π / 2) and the whole line.

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