Trigonometric and hyperbolic trigonometric functions

This file contains the definitions of the sine, cosine, tangent, hyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions.

def Complex.sin (z : ℂ) : ℂ

The complex sine function, defined via exp

Equations

• Complex.sin z = (Complex.exp (-z * Complex.I) - Complex.exp (z * Complex.I)) * Complex.I / 2

def Complex.cos (z : ℂ) : ℂ

The complex cosine function, defined via exp

Equations

• Complex.cos z = (Complex.exp (z * Complex.I) + Complex.exp (-z * Complex.I)) / 2

def Complex.tan (z : ℂ) : ℂ

The complex tangent function, defined as sin z / cos z

Equations

• Complex.tan z = Complex.sin z / Complex.cos z

def Complex.cot (z : ℂ) : ℂ

The complex cotangent function, defined as cos z / sin z

Equations

• z.cot = Complex.cos z / Complex.sin z

def Complex.sinh (z : ℂ) : ℂ

The complex hyperbolic sine function, defined via exp

Equations

• Complex.sinh z = (Complex.exp z - Complex.exp (-z)) / 2

def Complex.cosh (z : ℂ) : ℂ

The complex hyperbolic cosine function, defined via exp

Equations

• Complex.cosh z = (Complex.exp z + Complex.exp (-z)) / 2

def Complex.tanh (z : ℂ) : ℂ

The complex hyperbolic tangent function, defined as sinh z / cosh z

Equations

• Complex.tanh z = Complex.sinh z / Complex.cosh z

def Real.sin (x : ℝ) : ℝ

The real sine function, defined as the real part of the complex sine

Equations

• Real.sin x = (Complex.sin ↑x).re

def Real.cos (x : ℝ) : ℝ

The real cosine function, defined as the real part of the complex cosine

Equations

• Real.cos x = (Complex.cos ↑x).re

def Real.tan (x : ℝ) : ℝ

The real tangent function, defined as the real part of the complex tangent

Equations

• Real.tan x = (Complex.tan ↑x).re

def Real.cot (x : ℝ) : ℝ

The real cotangent function, defined as the real part of the complex cotangent

Equations

• x.cot = (↑x).cot.re

def Real.sinh (x : ℝ) : ℝ

The real hypebolic sine function, defined as the real part of the complex hyperbolic sine

Equations

• Real.sinh x = (Complex.sinh ↑x).re

def Real.cosh (x : ℝ) : ℝ

The real hypebolic cosine function, defined as the real part of the complex hyperbolic cosine

Equations

• Real.cosh x = (Complex.cosh ↑x).re

def Real.tanh (x : ℝ) : ℝ

The real hypebolic tangent function, defined as the real part of the complex hyperbolic tangent

Equations

• Real.tanh x = (Complex.tanh ↑x).re

theorem Complex.two_sinh (x : ℂ) : 2 * Complex.sinh x = Complex.exp x - Complex.exp (-x)

theorem Complex.two_cosh (x : ℂ) : 2 * Complex.cosh x = Complex.exp x + Complex.exp (-x)

@[simp]

theorem Complex.sinh_zero : Complex.sinh 0 = 0

@[simp]

theorem Complex.sinh_neg (x : ℂ) : Complex.sinh (-x) = -Complex.sinh x

theorem Complex.sinh_add (x y : ℂ) : Complex.sinh (x + y) = Complex.sinh x * Complex.cosh y + Complex.cosh x * Complex.sinh y

@[simp]

theorem Complex.cosh_zero : Complex.cosh 0 = 1

@[simp]

theorem Complex.cosh_neg (x : ℂ) : Complex.cosh (-x) = Complex.cosh x

theorem Complex.cosh_add (x y : ℂ) : Complex.cosh (x + y) = Complex.cosh x * Complex.cosh y + Complex.sinh x * Complex.sinh y

theorem Complex.sinh_sub (x y : ℂ) : Complex.sinh (x - y) = Complex.sinh x * Complex.cosh y - Complex.cosh x * Complex.sinh y

theorem Complex.cosh_sub (x y : ℂ) : Complex.cosh (x - y) = Complex.cosh x * Complex.cosh y - Complex.sinh x * Complex.sinh y

theorem Complex.sinh_conj (x : ℂ) : Complex.sinh ((starRingEnd ℂ) x) = (starRingEnd ℂ) (Complex.sinh x)

@[simp]

theorem Complex.ofReal_sinh_ofReal_re (x : ℝ) : ↑(Complex.sinh ↑x).re = Complex.sinh ↑x

@[simp]

theorem Complex.ofReal_sinh (x : ℝ) : ↑(Real.sinh x) = Complex.sinh ↑x

@[simp]

theorem Complex.sinh_ofReal_im (x : ℝ) : (Complex.sinh ↑x).im = 0

theorem Complex.sinh_ofReal_re (x : ℝ) : (Complex.sinh ↑x).re = Real.sinh x

theorem Complex.cosh_conj (x : ℂ) : Complex.cosh ((starRingEnd ℂ) x) = (starRingEnd ℂ) (Complex.cosh x)

theorem Complex.ofReal_cosh_ofReal_re (x : ℝ) : ↑(Complex.cosh ↑x).re = Complex.cosh ↑x

@[simp]

theorem Complex.ofReal_cosh (x : ℝ) : ↑(Real.cosh x) = Complex.cosh ↑x

@[simp]

theorem Complex.cosh_ofReal_im (x : ℝ) : (Complex.cosh ↑x).im = 0

@[simp]

theorem Complex.cosh_ofReal_re (x : ℝ) : (Complex.cosh ↑x).re = Real.cosh x

theorem Complex.tanh_eq_sinh_div_cosh (x : ℂ) : Complex.tanh x = Complex.sinh x / Complex.cosh x

@[simp]

theorem Complex.tanh_zero : Complex.tanh 0 = 0

@[simp]

theorem Complex.tanh_neg (x : ℂ) : Complex.tanh (-x) = -Complex.tanh x

theorem Complex.tanh_conj (x : ℂ) : Complex.tanh ((starRingEnd ℂ) x) = (starRingEnd ℂ) (Complex.tanh x)

@[simp]

theorem Complex.ofReal_tanh_ofReal_re (x : ℝ) : ↑(Complex.tanh ↑x).re = Complex.tanh ↑x

@[simp]

theorem Complex.ofReal_tanh (x : ℝ) : ↑(Real.tanh x) = Complex.tanh ↑x

@[simp]

theorem Complex.tanh_ofReal_im (x : ℝ) : (Complex.tanh ↑x).im = 0

theorem Complex.tanh_ofReal_re (x : ℝ) : (Complex.tanh ↑x).re = Real.tanh x

@[simp]

theorem Complex.cosh_add_sinh (x : ℂ) : Complex.cosh x + Complex.sinh x = Complex.exp x

@[simp]

theorem Complex.sinh_add_cosh (x : ℂ) : Complex.sinh x + Complex.cosh x = Complex.exp x

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theorem Complex.exp_sub_cosh (x : ℂ) : Complex.exp x - Complex.cosh x = Complex.sinh x

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theorem Complex.exp_sub_sinh (x : ℂ) : Complex.exp x - Complex.sinh x = Complex.cosh x

@[simp]

theorem Complex.cosh_sub_sinh (x : ℂ) : Complex.cosh x - Complex.sinh x = Complex.exp (-x)

@[simp]

theorem Complex.sinh_sub_cosh (x : ℂ) : Complex.sinh x - Complex.cosh x = -Complex.exp (-x)

@[simp]

theorem Complex.cosh_sq_sub_sinh_sq (x : ℂ) : Complex.cosh x ^ 2 - Complex.sinh x ^ 2 = 1

theorem Complex.cosh_sq (x : ℂ) : Complex.cosh x ^ 2 = Complex.sinh x ^ 2 + 1

theorem Complex.sinh_sq (x : ℂ) : Complex.sinh x ^ 2 = Complex.cosh x ^ 2 - 1

theorem Complex.cosh_two_mul (x : ℂ) : Complex.cosh (2 * x) = Complex.cosh x ^ 2 + Complex.sinh x ^ 2

theorem Complex.sinh_two_mul (x : ℂ) : Complex.sinh (2 * x) = 2 * Complex.sinh x * Complex.cosh x

theorem Complex.cosh_three_mul (x : ℂ) : Complex.cosh (3 * x) = 4 * Complex.cosh x ^ 3 - 3 * Complex.cosh x

theorem Complex.sinh_three_mul (x : ℂ) : Complex.sinh (3 * x) = 4 * Complex.sinh x ^ 3 + 3 * Complex.sinh x

@[simp]

theorem Complex.sin_zero : Complex.sin 0 = 0

@[simp]

theorem Complex.sin_neg (x : ℂ) : Complex.sin (-x) = -Complex.sin x

theorem Complex.two_sin (x : ℂ) : 2 * Complex.sin x = (Complex.exp (-x * Complex.I) - Complex.exp (x * Complex.I)) * Complex.I

theorem Complex.two_cos (x : ℂ) : 2 * Complex.cos x = Complex.exp (x * Complex.I) + Complex.exp (-x * Complex.I)

theorem Complex.sinh_mul_I (x : ℂ) : Complex.sinh (x * Complex.I) = Complex.sin x * Complex.I

theorem Complex.cosh_mul_I (x : ℂ) : Complex.cosh (x * Complex.I) = Complex.cos x

theorem Complex.tanh_mul_I (x : ℂ) : Complex.tanh (x * Complex.I) = Complex.tan x * Complex.I

theorem Complex.cos_mul_I (x : ℂ) : Complex.cos (x * Complex.I) = Complex.cosh x

theorem Complex.sin_mul_I (x : ℂ) : Complex.sin (x * Complex.I) = Complex.sinh x * Complex.I

theorem Complex.tan_mul_I (x : ℂ) : Complex.tan (x * Complex.I) = Complex.tanh x * Complex.I

theorem Complex.sin_add (x y : ℂ) : Complex.sin (x + y) = Complex.sin x * Complex.cos y + Complex.cos x * Complex.sin y

@[simp]

theorem Complex.cos_zero : Complex.cos 0 = 1

@[simp]

theorem Complex.cos_neg (x : ℂ) : Complex.cos (-x) = Complex.cos x

theorem Complex.cos_add (x y : ℂ) : Complex.cos (x + y) = Complex.cos x * Complex.cos y - Complex.sin x * Complex.sin y

theorem Complex.sin_sub (x y : ℂ) : Complex.sin (x - y) = Complex.sin x * Complex.cos y - Complex.cos x * Complex.sin y

theorem Complex.cos_sub (x y : ℂ) : Complex.cos (x - y) = Complex.cos x * Complex.cos y + Complex.sin x * Complex.sin y

theorem Complex.sin_add_mul_I (x y : ℂ) : Complex.sin (x + y * Complex.I) = Complex.sin x * Complex.cosh y + Complex.cos x * Complex.sinh y * Complex.I

theorem Complex.sin_eq (z : ℂ) : Complex.sin z = Complex.sin ↑z.re * Complex.cosh ↑z.im + Complex.cos ↑z.re * Complex.sinh ↑z.im * Complex.I

theorem Complex.cos_add_mul_I (x y : ℂ) : Complex.cos (x + y * Complex.I) = Complex.cos x * Complex.cosh y - Complex.sin x * Complex.sinh y * Complex.I

theorem Complex.cos_eq (z : ℂ) : Complex.cos z = Complex.cos ↑z.re * Complex.cosh ↑z.im - Complex.sin ↑z.re * Complex.sinh ↑z.im * Complex.I

theorem Complex.sin_sub_sin (x y : ℂ) : Complex.sin x - Complex.sin y = 2 * Complex.sin ((x - y) / 2) * Complex.cos ((x + y) / 2)

theorem Complex.cos_sub_cos (x y : ℂ) : Complex.cos x - Complex.cos y = -2 * Complex.sin ((x + y) / 2) * Complex.sin ((x - y) / 2)

theorem Complex.sin_add_sin (x y : ℂ) : Complex.sin x + Complex.sin y = 2 * Complex.sin ((x + y) / 2) * Complex.cos ((x - y) / 2)

theorem Complex.cos_add_cos (x y : ℂ) : Complex.cos x + Complex.cos y = 2 * Complex.cos ((x + y) / 2) * Complex.cos ((x - y) / 2)

theorem Complex.sin_conj (x : ℂ) : Complex.sin ((starRingEnd ℂ) x) = (starRingEnd ℂ) (Complex.sin x)

@[simp]

theorem Complex.ofReal_sin_ofReal_re (x : ℝ) : ↑(Complex.sin ↑x).re = Complex.sin ↑x

@[simp]

theorem Complex.ofReal_sin (x : ℝ) : ↑(Real.sin x) = Complex.sin ↑x

@[simp]

theorem Complex.sin_ofReal_im (x : ℝ) : (Complex.sin ↑x).im = 0

theorem Complex.sin_ofReal_re (x : ℝ) : (Complex.sin ↑x).re = Real.sin x

theorem Complex.cos_conj (x : ℂ) : Complex.cos ((starRingEnd ℂ) x) = (starRingEnd ℂ) (Complex.cos x)

@[simp]

theorem Complex.ofReal_cos_ofReal_re (x : ℝ) : ↑(Complex.cos ↑x).re = Complex.cos ↑x

@[simp]

theorem Complex.ofReal_cos (x : ℝ) : ↑(Real.cos x) = Complex.cos ↑x

@[simp]

theorem Complex.cos_ofReal_im (x : ℝ) : (Complex.cos ↑x).im = 0

theorem Complex.cos_ofReal_re (x : ℝ) : (Complex.cos ↑x).re = Real.cos x

@[simp]

theorem Complex.tan_zero : Complex.tan 0 = 0

theorem Complex.tan_eq_sin_div_cos (x : ℂ) : Complex.tan x = Complex.sin x / Complex.cos x

theorem Complex.cot_eq_cos_div_sin (x : ℂ) : x.cot = Complex.cos x / Complex.sin x

theorem Complex.tan_mul_cos {x : ℂ} (hx : Complex.cos x ≠ 0) : Complex.tan x * Complex.cos x = Complex.sin x

@[simp]

theorem Complex.tan_neg (x : ℂ) : Complex.tan (-x) = -Complex.tan x

theorem Complex.tan_conj (x : ℂ) : Complex.tan ((starRingEnd ℂ) x) = (starRingEnd ℂ) (Complex.tan x)

theorem Complex.cot_conj (x : ℂ) : ((starRingEnd ℂ) x).cot = (starRingEnd ℂ) x.cot

@[simp]

theorem Complex.ofReal_tan_ofReal_re (x : ℝ) : ↑(Complex.tan ↑x).re = Complex.tan ↑x

@[simp]

theorem Complex.ofReal_cot_ofReal_re (x : ℝ) : ↑(↑x).cot.re = (↑x).cot

@[simp]

theorem Complex.ofReal_tan (x : ℝ) : ↑(Real.tan x) = Complex.tan ↑x

@[simp]

theorem Complex.ofReal_cot (x : ℝ) : ↑x.cot = (↑x).cot

@[simp]

theorem Complex.tan_ofReal_im (x : ℝ) : (Complex.tan ↑x).im = 0

theorem Complex.tan_ofReal_re (x : ℝ) : (Complex.tan ↑x).re = Real.tan x

theorem Complex.cos_add_sin_I (x : ℂ) : Complex.cos x + Complex.sin x * Complex.I = Complex.exp (x * Complex.I)

theorem Complex.cos_sub_sin_I (x : ℂ) : Complex.cos x - Complex.sin x * Complex.I = Complex.exp (-x * Complex.I)

@[simp]

theorem Complex.sin_sq_add_cos_sq (x : ℂ) : Complex.sin x ^ 2 + Complex.cos x ^ 2 = 1

@[simp]

theorem Complex.cos_sq_add_sin_sq (x : ℂ) : Complex.cos x ^ 2 + Complex.sin x ^ 2 = 1

theorem Complex.cos_two_mul' (x : ℂ) : Complex.cos (2 * x) = Complex.cos x ^ 2 - Complex.sin x ^ 2

theorem Complex.cos_two_mul (x : ℂ) : Complex.cos (2 * x) = 2 * Complex.cos x ^ 2 - 1

theorem Complex.sin_two_mul (x : ℂ) : Complex.sin (2 * x) = 2 * Complex.sin x * Complex.cos x

theorem Complex.cos_sq (x : ℂ) : Complex.cos x ^ 2 = 1 / 2 + Complex.cos (2 * x) / 2

theorem Complex.cos_sq' (x : ℂ) : Complex.cos x ^ 2 = 1 - Complex.sin x ^ 2

theorem Complex.sin_sq (x : ℂ) : Complex.sin x ^ 2 = 1 - Complex.cos x ^ 2

theorem Complex.inv_one_add_tan_sq {x : ℂ} (hx : Complex.cos x ≠ 0) : (1 + Complex.tan x ^ 2)⁻¹ = Complex.cos x ^ 2

theorem Complex.tan_sq_div_one_add_tan_sq {x : ℂ} (hx : Complex.cos x ≠ 0) : Complex.tan x ^ 2 / (1 + Complex.tan x ^ 2) = Complex.sin x ^ 2

theorem Complex.cos_three_mul (x : ℂ) : Complex.cos (3 * x) = 4 * Complex.cos x ^ 3 - 3 * Complex.cos x

theorem Complex.sin_three_mul (x : ℂ) : Complex.sin (3 * x) = 3 * Complex.sin x - 4 * Complex.sin x ^ 3

theorem Complex.exp_mul_I (x : ℂ) : Complex.exp (x * Complex.I) = Complex.cos x + Complex.sin x * Complex.I

theorem Complex.exp_add_mul_I (x y : ℂ) : Complex.exp (x + y * Complex.I) = Complex.exp x * (Complex.cos y + Complex.sin y * Complex.I)

theorem Complex.exp_eq_exp_re_mul_sin_add_cos (x : ℂ) : Complex.exp x = Complex.exp ↑x.re * (Complex.cos ↑x.im + Complex.sin ↑x.im * Complex.I)

theorem Complex.exp_re (x : ℂ) : (Complex.exp x).re = Real.exp x.re * Real.cos x.im

theorem Complex.exp_im (x : ℂ) : (Complex.exp x).im = Real.exp x.re * Real.sin x.im

@[simp]

theorem Complex.exp_ofReal_mul_I_re (x : ℝ) : (Complex.exp (↑x * Complex.I)).re = Real.cos x

@[simp]

theorem Complex.exp_ofReal_mul_I_im (x : ℝ) : (Complex.exp (↑x * Complex.I)).im = Real.sin x

theorem Complex.cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) : (Complex.cos z + Complex.sin z * Complex.I) ^ n = Complex.cos (↑n * z) + Complex.sin (↑n * z) * Complex.I

De Moivre's formula

@[simp]

theorem Real.sin_zero : Real.sin 0 = 0

@[simp]

theorem Real.sin_neg (x : ℝ) : Real.sin (-x) = -Real.sin x

theorem Real.sin_add (x y : ℝ) : Real.sin (x + y) = Real.sin x * Real.cos y + Real.cos x * Real.sin y

@[simp]

theorem Real.cos_zero : Real.cos 0 = 1

@[simp]

theorem Real.cos_neg (x : ℝ) : Real.cos (-x) = Real.cos x

@[simp]

theorem Real.cos_abs (x : ℝ) : Real.cos |x| = Real.cos x

theorem Real.cos_add (x y : ℝ) : Real.cos (x + y) = Real.cos x * Real.cos y - Real.sin x * Real.sin y

theorem Real.sin_sub (x y : ℝ) : Real.sin (x - y) = Real.sin x * Real.cos y - Real.cos x * Real.sin y

theorem Real.cos_sub (x y : ℝ) : Real.cos (x - y) = Real.cos x * Real.cos y + Real.sin x * Real.sin y

theorem Real.sin_sub_sin (x y : ℝ) : Real.sin x - Real.sin y = 2 * Real.sin ((x - y) / 2) * Real.cos ((x + y) / 2)

theorem Real.cos_sub_cos (x y : ℝ) : Real.cos x - Real.cos y = -2 * Real.sin ((x + y) / 2) * Real.sin ((x - y) / 2)

theorem Real.cos_add_cos (x y : ℝ) : Real.cos x + Real.cos y = 2 * Real.cos ((x + y) / 2) * Real.cos ((x - y) / 2)

theorem Real.two_mul_sin_mul_sin (x y : ℝ) : 2 * Real.sin x * Real.sin y = Real.cos (x - y) - Real.cos (x + y)

theorem Real.two_mul_cos_mul_cos (x y : ℝ) : 2 * Real.cos x * Real.cos y = Real.cos (x - y) + Real.cos (x + y)

theorem Real.two_mul_sin_mul_cos (x y : ℝ) : 2 * Real.sin x * Real.cos y = Real.sin (x - y) + Real.sin (x + y)

theorem Real.tan_eq_sin_div_cos (x : ℝ) : Real.tan x = Real.sin x / Real.cos x

theorem Real.cot_eq_cos_div_sin (x : ℝ) : x.cot = Real.cos x / Real.sin x

theorem Real.tan_mul_cos {x : ℝ} (hx : Real.cos x ≠ 0) : Real.tan x * Real.cos x = Real.sin x

@[simp]

theorem Real.tan_zero : Real.tan 0 = 0

@[simp]

theorem Real.tan_neg (x : ℝ) : Real.tan (-x) = -Real.tan x

@[simp]

theorem Real.sin_sq_add_cos_sq (x : ℝ) : Real.sin x ^ 2 + Real.cos x ^ 2 = 1

@[simp]

theorem Real.cos_sq_add_sin_sq (x : ℝ) : Real.cos x ^ 2 + Real.sin x ^ 2 = 1

theorem Real.sin_sq_le_one (x : ℝ) : Real.sin x ^ 2 ≤ 1

theorem Real.cos_sq_le_one (x : ℝ) : Real.cos x ^ 2 ≤ 1

theorem Real.abs_sin_le_one (x : ℝ) : |Real.sin x| ≤ 1

theorem Real.abs_cos_le_one (x : ℝ) : |Real.cos x| ≤ 1

theorem Real.sin_le_one (x : ℝ) : Real.sin x ≤ 1

theorem Real.cos_le_one (x : ℝ) : Real.cos x ≤ 1

theorem Real.neg_one_le_sin (x : ℝ) : -1 ≤ Real.sin x

theorem Real.neg_one_le_cos (x : ℝ) : -1 ≤ Real.cos x

theorem Real.cos_two_mul (x : ℝ) : Real.cos (2 * x) = 2 * Real.cos x ^ 2 - 1

theorem Real.cos_two_mul' (x : ℝ) : Real.cos (2 * x) = Real.cos x ^ 2 - Real.sin x ^ 2

theorem Real.sin_two_mul (x : ℝ) : Real.sin (2 * x) = 2 * Real.sin x * Real.cos x

theorem Real.cos_sq (x : ℝ) : Real.cos x ^ 2 = 1 / 2 + Real.cos (2 * x) / 2

theorem Real.cos_sq' (x : ℝ) : Real.cos x ^ 2 = 1 - Real.sin x ^ 2

theorem Real.sin_sq (x : ℝ) : Real.sin x ^ 2 = 1 - Real.cos x ^ 2

theorem Real.sin_sq_eq_half_sub (x : ℝ) : Real.sin x ^ 2 = 1 / 2 - Real.cos (2 * x) / 2

theorem Real.abs_sin_eq_sqrt_one_sub_cos_sq (x : ℝ) : |Real.sin x| = √(1 - Real.cos x ^ 2)

theorem Real.abs_cos_eq_sqrt_one_sub_sin_sq (x : ℝ) : |Real.cos x| = √(1 - Real.sin x ^ 2)

theorem Real.inv_one_add_tan_sq {x : ℝ} (hx : Real.cos x ≠ 0) : (1 + Real.tan x ^ 2)⁻¹ = Real.cos x ^ 2

theorem Real.tan_sq_div_one_add_tan_sq {x : ℝ} (hx : Real.cos x ≠ 0) : Real.tan x ^ 2 / (1 + Real.tan x ^ 2) = Real.sin x ^ 2

theorem Real.inv_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < Real.cos x) : (√(1 + Real.tan x ^ 2))⁻¹ = Real.cos x

theorem Real.tan_div_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < Real.cos x) : Real.tan x / √(1 + Real.tan x ^ 2) = Real.sin x

theorem Real.cos_three_mul (x : ℝ) : Real.cos (3 * x) = 4 * Real.cos x ^ 3 - 3 * Real.cos x

theorem Real.sin_three_mul (x : ℝ) : Real.sin (3 * x) = 3 * Real.sin x - 4 * Real.sin x ^ 3

theorem Real.sinh_eq (x : ℝ) : Real.sinh x = (Real.exp x - Real.exp (-x)) / 2

The definition of sinh in terms of exp.

@[simp]

theorem Real.sinh_zero : Real.sinh 0 = 0

@[simp]

theorem Real.sinh_neg (x : ℝ) : Real.sinh (-x) = -Real.sinh x

theorem Real.sinh_add (x y : ℝ) : Real.sinh (x + y) = Real.sinh x * Real.cosh y + Real.cosh x * Real.sinh y

theorem Real.cosh_eq (x : ℝ) : Real.cosh x = (Real.exp x + Real.exp (-x)) / 2

The definition of cosh in terms of exp.

@[simp]

theorem Real.cosh_zero : Real.cosh 0 = 1

@[simp]

theorem Real.cosh_neg (x : ℝ) : Real.cosh (-x) = Real.cosh x

@[simp]

theorem Real.cosh_abs (x : ℝ) : Real.cosh |x| = Real.cosh x

theorem Real.cosh_add (x y : ℝ) : Real.cosh (x + y) = Real.cosh x * Real.cosh y + Real.sinh x * Real.sinh y

theorem Real.sinh_sub (x y : ℝ) : Real.sinh (x - y) = Real.sinh x * Real.cosh y - Real.cosh x * Real.sinh y

theorem Real.cosh_sub (x y : ℝ) : Real.cosh (x - y) = Real.cosh x * Real.cosh y - Real.sinh x * Real.sinh y

theorem Real.tanh_eq_sinh_div_cosh (x : ℝ) : Real.tanh x = Real.sinh x / Real.cosh x

@[simp]

theorem Real.tanh_zero : Real.tanh 0 = 0

@[simp]

theorem Real.tanh_neg (x : ℝ) : Real.tanh (-x) = -Real.tanh x

@[simp]

theorem Real.cosh_add_sinh (x : ℝ) : Real.cosh x + Real.sinh x = Real.exp x

@[simp]

theorem Real.sinh_add_cosh (x : ℝ) : Real.sinh x + Real.cosh x = Real.exp x

@[simp]

theorem Real.exp_sub_cosh (x : ℝ) : Real.exp x - Real.cosh x = Real.sinh x

@[simp]

theorem Real.exp_sub_sinh (x : ℝ) : Real.exp x - Real.sinh x = Real.cosh x

@[simp]

theorem Real.cosh_sub_sinh (x : ℝ) : Real.cosh x - Real.sinh x = Real.exp (-x)

@[simp]

theorem Real.sinh_sub_cosh (x : ℝ) : Real.sinh x - Real.cosh x = -Real.exp (-x)

@[simp]

theorem Real.cosh_sq_sub_sinh_sq (x : ℝ) : Real.cosh x ^ 2 - Real.sinh x ^ 2 = 1

theorem Real.cosh_sq (x : ℝ) : Real.cosh x ^ 2 = Real.sinh x ^ 2 + 1

theorem Real.cosh_sq' (x : ℝ) : Real.cosh x ^ 2 = 1 + Real.sinh x ^ 2

theorem Real.sinh_sq (x : ℝ) : Real.sinh x ^ 2 = Real.cosh x ^ 2 - 1

theorem Real.cosh_two_mul (x : ℝ) : Real.cosh (2 * x) = Real.cosh x ^ 2 + Real.sinh x ^ 2

theorem Real.sinh_two_mul (x : ℝ) : Real.sinh (2 * x) = 2 * Real.sinh x * Real.cosh x

theorem Real.cosh_three_mul (x : ℝ) : Real.cosh (3 * x) = 4 * Real.cosh x ^ 3 - 3 * Real.cosh x

theorem Real.sinh_three_mul (x : ℝ) : Real.sinh (3 * x) = 4 * Real.sinh x ^ 3 + 3 * Real.sinh x

theorem Real.cosh_pos (x : ℝ) : 0 < Real.cosh x

Real.cosh is always positive

theorem Real.sinh_lt_cosh (x : ℝ) : Real.sinh x < Real.cosh x

theorem Real.cos_bound {x : ℝ} (hx : |x| ≤ 1) : |Real.cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96)

theorem Real.sin_bound {x : ℝ} (hx : |x| ≤ 1) : |Real.sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96)

theorem Real.cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < Real.cos x

theorem Real.sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < Real.sin x

theorem Real.sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < Real.sin x

theorem Real.cos_one_le : Real.cos 1 ≤ 2 / 3

theorem Real.cos_one_pos : 0 < Real.cos 1

theorem Real.cos_two_neg : Real.cos 2 < 0

def Mathlib.Meta.Positivity.evalCosh : Mathlib.Meta.Positivity.PositivityExt

Extension for the positivity tactic: Real.cosh is always positive.

@[simp]

theorem Complex.abs_cos_add_sin_mul_I (x : ℝ) : Complex.abs (Complex.cos ↑x + Complex.sin ↑x * Complex.I) = 1

@[simp]

theorem Complex.abs_exp_ofReal_mul_I (x : ℝ) : Complex.abs (Complex.exp (↑x * Complex.I)) = 1

theorem Complex.abs_exp (z : ℂ) : Complex.abs (Complex.exp z) = Real.exp z.re

theorem Complex.abs_exp_eq_iff_re_eq {x y : ℂ} : Complex.abs (Complex.exp x) = Complex.abs (Complex.exp y) ↔ x.re = y.re