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 : ℝ} {y : ℝ} (h : ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * Real.pi / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * Real.pi / 2) ∨ (∃ (k : ℤ), x = (2 * ↑k + 1) * Real.pi / 2) ∧ ∃ (l : ℤ), y = (2 * ↑l + 1) * Real.pi / 2) : (x + y).tan = (x.tan + y.tan) / (1 - x.tan * y.tan)

theorem Real.tan_add' {x : ℝ} {y : ℝ} (h : (∀ (k : ℤ), x ≠ (2 * ↑k + 1) * Real.pi / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * Real.pi / 2) : (x + y).tan = (x.tan + y.tan) / (1 - x.tan * y.tan)

theorem Real.tan_two_mul {x : ℝ} : (2 * x).tan = 2 * x.tan / (1 - x.tan ^ 2)

theorem Real.tan_int_mul_pi_div_two (n : ℤ) : (↑n * Real.pi / 2).tan = 0

theorem Real.continuousOn_tan : ContinuousOn Real.tan {x : ℝ | x.cos ≠ 0}

theorem Real.continuous_tan : Continuous fun (x : ↑{x : ℝ | x.cos ≠ 0}) => (↑x).tan

theorem Real.continuousOn_tan_Ioo : ContinuousOn Real.tan (Set.Ioo (-(Real.pi / 2)) (Real.pi / 2))

theorem Real.surjOn_tan : Set.SurjOn Real.tan (Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)) Set.univ

theorem Real.tan_surjective : Function.Surjective Real.tan

theorem Real.image_tan_Ioo : Real.tan '' Set.Ioo (-(Real.pi / 2)) (Real.pi / 2) = Set.univ

def Real.tanOrderIso : ↑(Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)) ≃o ℝ

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

Equations

• One or more equations did not get rendered due to their size.

noncomputable def Real.arctan (x : ℝ) : ℝ

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

Equations

• x.arctan = ↑(Real.tanOrderIso.symm x)

@[simp]

theorem Real.tan_arctan (x : ℝ) : x.arctan.tan = x

theorem Real.arctan_mem_Ioo (x : ℝ) : x.arctan ∈ Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)

@[simp]

theorem Real.range_arctan : Set.range Real.arctan = Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)

theorem Real.arctan_tan {x : ℝ} (hx₁ : -(Real.pi / 2) < x) (hx₂ : x < Real.pi / 2) : x.tan.arctan = x

theorem Real.cos_arctan_pos (x : ℝ) : 0 < x.arctan.cos

theorem Real.cos_sq_arctan (x : ℝ) : x.arctan.cos ^ 2 = 1 / (1 + x ^ 2)

theorem Real.sin_arctan (x : ℝ) : x.arctan.sin = x / √(1 + x ^ 2)

theorem Real.cos_arctan (x : ℝ) : x.arctan.cos = 1 / √(1 + x ^ 2)

theorem Real.arctan_lt_pi_div_two (x : ℝ) : x.arctan < Real.pi / 2

theorem Real.neg_pi_div_two_lt_arctan (x : ℝ) : -(Real.pi / 2) < x.arctan

theorem Real.arctan_eq_arcsin (x : ℝ) : x.arctan = (x / √(1 + x ^ 2)).arcsin

theorem Real.arcsin_eq_arctan {x : ℝ} (h : x ∈ Set.Ioo (-1) 1) : x.arcsin = (x / √(1 - x ^ 2)).arctan

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theorem Real.arctan_zero : Real.arctan 0 = 0

theorem Real.arctan_strictMono : StrictMono Real.arctan

theorem Real.arctan_injective : Function.Injective Real.arctan

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theorem Real.arctan_eq_zero_iff {x : ℝ} : x.arctan = 0 ↔ x = 0

theorem Real.tendsto_arctan_atTop : Filter.Tendsto Real.arctan Filter.atTop (nhdsWithin (Real.pi / 2) (Set.Iio (Real.pi / 2)))

theorem Real.tendsto_arctan_atBot : Filter.Tendsto Real.arctan Filter.atBot (nhdsWithin (-(Real.pi / 2)) (Set.Ioi (-(Real.pi / 2))))

theorem Real.arctan_eq_of_tan_eq {x : ℝ} {y : ℝ} (h : x.tan = y) (hx : x ∈ Set.Ioo (-(Real.pi / 2)) (Real.pi / 2)) : y.arctan = x

@[simp]

theorem Real.arctan_one : Real.arctan 1 = Real.pi / 4

@[simp]

theorem Real.arctan_neg (x : ℝ) : (-x).arctan = -x.arctan

theorem Real.arctan_eq_arccos {x : ℝ} (h : 0 ≤ x) : x.arctan = (√(1 + x ^ 2))⁻¹.arccos

theorem Real.arccos_eq_arctan {x : ℝ} (h : 0 < x) : x.arccos = (√(1 - x ^ 2) / x).arctan

theorem Real.arctan_inv_of_pos {x : ℝ} (h : 0 < x) : x⁻¹.arctan = Real.pi / 2 - x.arctan

theorem Real.arctan_inv_of_neg {x : ℝ} (h : x < 0) : x⁻¹.arctan = -(Real.pi / 2) - x.arctan

theorem Real.arctan_ne_mul_pi_div_two {x : ℝ} (k : ℤ) : x.arctan ≠ (2 * ↑k + 1) * Real.pi / 2

theorem Real.arctan_add_arctan_lt_pi_div_two {x : ℝ} {y : ℝ} (h : x * y < 1) : x.arctan + y.arctan < Real.pi / 2

theorem Real.arctan_add {x : ℝ} {y : ℝ} (h : x * y < 1) : x.arctan + y.arctan = ((x + y) / (1 - x * y)).arctan

theorem Real.arctan_add_eq_add_pi {x : ℝ} {y : ℝ} (h : 1 < x * y) (hx : 0 < x) : x.arctan + y.arctan = ((x + y) / (1 - x * y)).arctan + Real.pi

theorem Real.arctan_add_eq_sub_pi {x : ℝ} {y : ℝ} (h : 1 < x * y) (hx : x < 0) : x.arctan + y.arctan = ((x + y) / (1 - x * y)).arctan - Real.pi

theorem Real.two_mul_arctan {x : ℝ} (h₁ : -1 < x) (h₂ : x < 1) : 2 * x.arctan = (2 * x / (1 - x ^ 2)).arctan

theorem Real.two_mul_arctan_add_pi {x : ℝ} (h : 1 < x) : 2 * x.arctan = (2 * x / (1 - x ^ 2)).arctan + Real.pi

theorem Real.two_mul_arctan_sub_pi {x : ℝ} (h : x < -1) : 2 * x.arctan = (2 * x / (1 - x ^ 2)).arctan - Real.pi

theorem Real.arctan_inv_2_add_arctan_inv_3 : 2⁻¹.arctan + 3⁻¹.arctan = Real.pi / 4

theorem Real.two_mul_arctan_inv_2_sub_arctan_inv_7 : 2 * 2⁻¹.arctan - 7⁻¹.arctan = Real.pi / 4

theorem Real.two_mul_arctan_inv_3_add_arctan_inv_7 : 2 * 3⁻¹.arctan + 7⁻¹.arctan = Real.pi / 4

theorem Real.four_mul_arctan_inv_5_sub_arctan_inv_239 : 4 * 5⁻¹.arctan - 239⁻¹.arctan = Real.pi / 4

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

theorem Real.continuous_arctan : Continuous Real.arctan

theorem Real.continuousAt_arctan {x : ℝ} : ContinuousAt Real.arctan x

def Real.tanPartialHomeomorph : PartialHomeomorph ℝ ℝ

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

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Real.coe_tanPartialHomeomorph : ↑Real.tanPartialHomeomorph = Real.tan

@[simp]

theorem Real.coe_tanPartialHomeomorph_symm : ↑Real.tanPartialHomeomorph.symm = Real.arctan