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analysis.special_functions.trigonometric.bounds

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.

Tags #

sin, cos, tan, angle

theorem real.sin_lt {x : } (h : 0 < x) :
< x

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

theorem real.sin_gt_sub_cube {x : } (h : 0 < x) (h' : x 1) :
x - x ^ 3 / 4 <

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 : ) (h : 0) :
deriv (λ (y : ), - y) x = 1 / ^ 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 : ) (h1 : 0 < x) (h2 : x < ) :
x <

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.