Polynomial bounds for trigonometric functions

Main statements

This file contains upper and lower bounds for real trigonometric functions in terms of polynomials. See Trigonometric.Basic for more elementary inequalities, establishing the ranges of these functions, and their monotonicity in suitable intervals.

Here we prove the following:

• sin_lt: for x > 0 we have sin x < x.
• sin_gt_sub_cube: For 0 < x ≤ 1 we have x - x ^ 3 / 4 < sin x.
• lt_tan: for 0 < x < π/2 we have x < tan x.
• cos_le_one_div_sqrt_sq_add_one and cos_lt_one_div_sqrt_sq_add_one: for -3 * π / 2 ≤ x ≤ 3 * π / 2, we have cos x ≤ 1 / sqrt (x ^ 2 + 1), with strict inequality if x ≠ 0. (This bound is not quite optimal, but not far off)

Tags

sin, cos, tan, angle

theorem Real.sin_lt {x : ℝ} (h : 0 < x) : x.sin < x

For 0 < x, we have sin x < x.

theorem Real.sin_le {x : ℝ} (hx : 0 ≤ x) : x.sin ≤ x

theorem Real.lt_sin {x : ℝ} (hx : x < 0) : x < x.sin

theorem Real.le_sin {x : ℝ} (hx : x ≤ 0) : x ≤ x.sin

theorem Real.one_sub_sq_div_two_le_cos {x : ℝ} : 1 - x ^ 2 / 2 ≤ x.cos

theorem Real.two_div_pi_mul_le_sin {x : ℝ} (hx₀ : 0 ≤ x) (hx : x ≤ Real.pi / 2) : 2 / Real.pi * x ≤ x.sin

Jordan's inequality.

theorem Real.sin_le_two_div_pi_mul {x : ℝ} (hx : -(Real.pi / 2) ≤ x) (hx₀ : x ≤ 0) : x.sin ≤ 2 / Real.pi * x

Jordan's inequality for negative values.

theorem Real.one_sub_two_div_pi_mul_le_cos {x : ℝ} (hx₀ : 0 ≤ x) (hx : x ≤ Real.pi / 2) : 1 - 2 / Real.pi * x ≤ x.cos

Jordan's inequality for cos.

theorem Real.cos_quadratic_upper_bound {x : ℝ} (hx : |x| ≤ Real.pi) : x.cos ≤ 1 - 2 / Real.pi ^ 2 * x ^ 2

theorem Real.sin_gt_sub_cube {x : ℝ} (h : 0 < x) (h' : x ≤ 1) : x - x ^ 3 / 4 < x.sin

For 0 < x ≤ 1 we have x - x ^ 3 / 4 < sin x.

This is also true for x > 1, but it's nontrivial for x just above 1. This inequality is not tight; the tighter inequality is sin x > x - x ^ 3 / 6 for all x > 0, but this inequality has a simpler proof.

theorem Real.deriv_tan_sub_id (x : ℝ) (h : x.cos ≠ 0) : deriv (fun (y : ℝ) => y.tan - y) x = 1 / x.cos ^ 2 - 1

The derivative of tan x - x is 1/(cos x)^2 - 1 away from the zeroes of cos.

theorem Real.lt_tan {x : ℝ} (h1 : 0 < x) (h2 : x < Real.pi / 2) : x < x.tan

For all 0 < x < π/2 we have x < tan x.

This is proved by checking that the function tan x - x vanishes at zero and has non-negative derivative.

theorem Real.le_tan {x : ℝ} (h1 : 0 ≤ x) (h2 : x < Real.pi / 2) : x ≤ x.tan

theorem Real.cos_lt_one_div_sqrt_sq_add_one {x : ℝ} (hx1 : -(3 * Real.pi / 2) ≤ x) (hx2 : x ≤ 3 * Real.pi / 2) (hx3 : x ≠ 0) : x.cos < 1 / (x ^ 2 + 1).sqrt

theorem Real.cos_le_one_div_sqrt_sq_add_one {x : ℝ} (hx1 : -(3 * Real.pi / 2) ≤ x) (hx2 : x ≤ 3 * Real.pi / 2) : x.cos ≤ 1 / (x ^ 2 + 1).sqrt