Basic Trigonometric Inequalities

Unknown variable (angle): $$x$$
Set of integers: $$\mathbb{Z}$$
Integer: $$n$$
Set of real numbers: $$\mathbb{R}$$
Real number: $$a$$

Trigonometric functions: $$\sin x,$$ $$\cos x,$$ $$\tan x,$$ $$\cot x$$
Inverse trigonometric functions: $$\arcsin a,$$ $$\arccos a,$$ $$\arctan a,$$ $$\text {arccot } a$$

An inequality involving trigonometric functions of an unknown angle is called a trigonometric inequality.

Basic trigonometric equations have the form

The following $$16$$ inequalities refer to basic trigonometric inequalities:

$\sin x \gt a,\; \sin x \ge a,\; \sin x \lt a,\; \sin x \le a,$
$\cos x \gt a,\; \cos x \ge a,\; \cos x \lt a,\; \cos x \le a,$
$\tan x \gt a,\; \tan x \ge a,\; \tan x \lt a,\; \tan x \le a,$
$\cot x \gt a,\; \cot x \ge a,\; \cot x \lt a,\; \cot x \le a.$

Here $$x$$ is an unknown variable, $$a$$ can be any real number.

Inequalities of the form $$\sin x \gt a,$$ $$\sin x \ge a,$$ $$\sin x \lt a,$$ $$\sin x \le a$$

Inequality $$\sin x \gt a$$

If $$\left| a \right| \ge 1$$, the inequality $$\sin x \gt a$$ has no solutions: $$x \in \varnothing.$$

If $$a \lt -1$$, the solution of the inequality $$\sin x \gt a$$ is any real number: $$x \in \mathbb{R}.$$

For $$-1 \le a \lt 1$$, the solution of the inequality $$\sin x \gt a$$ is expressed in the form

$\arcsin a + 2\pi n \lt x \lt \pi - \arcsin a + 2\pi n, \;n \in \mathbb{Z}.$

Inequality $$\sin x \ge a$$

If $$a \gt 1$$, the inequality $$\sin x \ge a$$ has no solutions: $$x \in \varnothing.$$

If $$a \le -1$$, the solution of the inequality $$\sin x \ge a$$ is any real number: $$x \in \mathbb{R}.$$

Case $$a = 1:$$

$x = \pi/2 +2\pi n,\; n \in \mathbb{Z}.$

For $$-1 \lt a \lt 1$$, the solution of the non-strict inequality $$\sin x \ge a$$ includes the boundary angles and has the form

$\arcsin a + 2\pi n \le x \le \pi - \arcsin a + 2\pi n,\;n \in \mathbb{Z}.$

Inequality $$\sin x \lt a$$

If $$a \gt 1$$, the solution of the inequality $$\sin x \lt a$$ is any real number: $$x \in \mathbb{R}.$$

If $$a \le -1$$, the inequality $$\sin x \lt a$$ has no solutions: $$x \in \varnothing.$$

For $$-1 \lt a \le 1$$, the solution of the inequality $$\sin x \lt a$$ lies in the interval

$-\pi - \arcsin a + 2\pi n \lt x \lt \arcsin a + 2\pi n,\;n \in \mathbb{Z}.$

Inequality $$\sin x \le a$$

If $$a \ge 1$$, the solution of the inequality $$\sin x \le a$$ is any real number: $$x \in \mathbb{R}.$$

If $$a \lt -1$$, the inequality $$\sin x \le a$$ has no solutions: $$x \in \varnothing.$$

Case $$a = -1:$$

$x = -\pi/2 + 2\pi n,\;n \in \mathbb{Z}.$

For $$-1 \lt a \lt 1$$, the solution of the non-strict inequality $$\sin x \le a$$ is in the interval

$-\pi - \arcsin a + 2\pi n \le x \le \arcsin a + 2\pi n,\;n \in \mathbb{Z}.$

Inequalities of the form $$\cos x \gt a,$$ $$\cos x \ge a,$$ $$\cos x \lt a,$$ $$\cos x \le a$$

Inequality $$\cos x \gt a$$

If $$a \ge 1$$, the inequality $$\cos x \gt a$$ has no solutions: $$x \in \varnothing.$$

If $$a \lt -1$$, the solution of the inequality $$\cos x \gt a$$ is any real number: $$x \in \mathbb{R}.$$

For $$-1 \le a \lt 1$$, the solution of the inequality $$\cos x \gt a$$ has the form

$-\arccos a + 2\pi n \lt x \lt \arccos a + 2\pi n,\;n \in \mathbb{Z}.$

Inequality $$\cos x \ge a$$

If $$a \gt 1$$, the inequality $$\cos x \ge a$$ has no solutions: $$x \in \varnothing.$$

If $$a \le -1$$, the solution of the inequality $$\cos x \ge a$$ is any real number: $$x \in \mathbb{R}.$$

Case $$a = 1:$$

$x = 2\pi n,\;n \in \mathbb{Z}.$

For $$-1 \lt a \lt 1$$, the solution of the non-strict inequality $$\cos x \ge a$$ is expressed by the formula

$-\arccos a + 2\pi n \le x \le \arccos a + 2\pi n,\;n \in \mathbb{Z}.$

Inequality $$\cos x \lt a$$

If $$a \gt 1$$, the inequality $$\cos x \lt a$$ is true for any real value of $$x$$: $$x \in \mathbb{R}.$$

If $$a \le -1$$, the inequality $$\cos x \lt a$$ has no solutions: $$x \in \varnothing.$$

For $$-1 \lt a \le 1$$, the solution of the inequality $$\cos x \lt a$$ is written in the form

$\arccos a + 2\pi n \lt x \lt 2\pi - \arccos a + 2\pi n,\;n \in \mathbb{Z}.$

Inequality $$\cos x \le a$$

If $$a \ge 1$$, the solution of the inequality $$\cos x \le a$$ is any real number: $$x \in \mathbb{R}.$$

If $$a \lt -1$$, the inequality $$\cos x \le a$$ has no solutions: $$x \in \varnothing.$$

Case $$a = -1:$$

$x = \pi + 2\pi n,\;n \in \mathbb{Z}.$

For $$-1 \lt a \lt 1$$, the solution of the non-strict inequality $$\cos x \le a$$ is written as

$\arccos a + 2\pi n \le x \le 2\pi - \arccos a + 2\pi n,\;n \in \mathbb{Z}.$

Inequalities of the form $$\tan x \gt a,$$ $$\tan x \ge a,$$ $$\tan x \lt a,$$ $$\tan x \le a$$

Inequality $$\tan x \gt a$$

For any real value of $$a$$, the solution of the strict inequality $$\tan x \gt a$$ has the form

$\arctan a + \pi n \lt x \lt \pi/2 + \pi n,\;n \in \mathbb{Z}.$

Inequality $$\tan x \ge a$$

For any real value of $$a$$, the solution of the inequality $$\tan x \ge a$$ is expressed in the form

$\arctan a + \pi n \le x \lt \pi/2 + \pi n,\;n \in \mathbb{Z}.$

Inequality $$\tan x \lt a$$

For any value of $$a$$, the solution of the inequality $$\tan x \lt a$$ is written in the form

$-\pi/2 + \pi n \lt x \lt \arctan a + \pi n,\;n \in \mathbb{Z}.$

Inequality $$\tan x \le a$$

For any value of $$a$$, the inequality $$\tan x \le a$$ has the following solution:

$-\pi/2 + \pi n \lt x \le \arctan a + \pi n,\;n \in \mathbb{Z}.$

Inequalities of the form $$\cot x \gt a,$$ $$\cot x \ge a,$$ $$\cot x \lt a,$$ $$\cot x \le a$$

Inequality $$\cot x \gt a$$

For any value of $$a$$, the solution of the inequality $$\cot x \gt a$$ has the form

$\pi n \lt x \lt \text {arccot } a + \pi n,\;n \in \mathbb{Z}.$

Inequality $$\cot x \ge a$$

The non-strict inequality $$\cot x \ge a$$ has the similar solution:

$\pi n \lt x \le \text {arccot } a + \pi n,\;n \in \mathbb{Z}.$

Inequality $$\cot x \lt a$$

For any value of $$a$$, the solution of the inequality $$\cot x \lt a$$ lies on the open interval

$\text {arccot } a + \pi n \lt x \lt \pi + \pi n,\;n \in \mathbb{Z}.$

Inequality $$\cot x \le a$$

For any value of $$a$$, the solution of the non-strict inequality $$\cot x \le a$$ is in the half-open interval

$\text {arccot } a + \pi n \le x \lt \pi + \pi n,\;n \in \mathbb{Z}.$