# Formulas and Tables

Trigonometry# Basic Trigonometric Inequalities

Set of integers: \(\mathbb{Z}\)

Integer: \(n\)

Set of real numbers: \(\mathbb{R}\)

Real number: \(a\)

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.
- 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 \(\sin x \gt a,\) \(\sin x \ge a,\) \(\sin x \lt a,\) \(\sin x \le a\)

Figure 1.

Figure 2.

#### Inequality \(\sin x \gt a\)

- If \(\left| a \right| \ge 1\), the inequality \(\sin x \gt a\) has no solutions: \(x \in \emptyset\)
- 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

\(\arcsin a + 2\pi n \lt x\) \(\lt \pi – \arcsin a + 2\pi n,\) \(n \in \mathbb{Z}\) (Fig.1).

#### Inequality \(\sin x \ge a\)

- If \(a \gt 1\), the inequality \(\sin x \ge a\) has no solutions: \(x \in \emptyset\)
- 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

\(\arcsin a + 2\pi n \le x \) \(\le \pi – \arcsin a + 2\pi n,\) \(n \in \mathbb{Z}\) (Fig.1).

#### 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: \(x \in \emptyset\)
- For \(-1 \lt a \le 1\), the solution of the inequality \(\sin x \lt a\) lies in the interval

\(-\pi – \arcsin a + 2\pi n \lt x \) \(\lt \arcsin a + 2\pi n,\) \(n \in \mathbb{Z}\) (Fig.2).

#### 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 \(a < -1\), the inequality \(\sin x \le a\) has no solutions: \(x \in \emptyset\)
- 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

\(-\pi – \arcsin a + 2\pi n \le x\) \(\le \arcsin a + 2\pi n,\) \(n \in \mathbb{Z}\) (Fig.2).

### Inequalities of the form \(\cos x \gt a,\) \(\cos x \ge a,\) \(\cos x \lt a,\) \(\cos x \le a\)

Figure 3.

Figure 4.

#### Inequality \(\cos x \gt a\)

- If \(a \ge 1\), the inequality \(\cos x \gt a\) has no solutions: \(x \in \emptyset\)
- 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

\(-\arccos a + 2\pi n \lt x\) \(\lt \arccos a + 2\pi n,\) \(n \in \mathbb{Z}\) (Fig.3).

#### Inequality \(\cos x \ge a\)

- If \(a \gt 1\), the inequality \(\cos x \ge a\) has no solutions: \(x \in \emptyset\)
- 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

\(-\arccos a + 2\pi n \le x\) \(\le \arccos a + 2\pi n,\) \(n \in \mathbb{Z}\) (Fig.3).

#### 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: \(x \in \emptyset\)
- For \(-1 < a \le 1\), the solution of the inequality \(\cos x \lt a\) is written in the form \(\arccos a + 2\pi n \lt x\) \(\lt 2\pi - \arccos a + 2\pi n,\) \(n \in \mathbb{Z}\) (Fig.4).

#### 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: \(x \in \emptyset\)
- 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

\(\arccos a + 2\pi n \le x\) \(\le 2\pi – \arccos a + 2\pi n,\) \(n \in \mathbb{Z}\) (Fig.4).

### Inequalities of the form \(\tan x \gt a,\) \(\tan x \ge a,\) \(\tan x \lt a,\) \(\tan x \le a\)

Figure 5.

Figure 6.

#### Inequality \(\tan x \gt a\)

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

\(\arctan a + \pi n \lt x\) \(\lt \pi/2 + \pi n,\) \(n \in \mathbb{Z}\) (Fig.5).

#### 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

\(\arctan a + \pi n \le x\) \(\lt \pi/2 + \pi n,\) \(n \in \mathbb{Z}\) (Fig.5).

#### Inequality \(\tan x \lt a\)

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

\(-\pi/2 + \pi n \lt x\) \(\lt \arctan a + \pi n,\) \(n \in \mathbb{Z}\) (Fig.6).

#### Inequality \(\tan x \le a\)

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

\(-\pi/2 + \pi n \lt x\) \(\le \arctan a + \pi n,\) \(n \in \mathbb{Z}\) (Fig.6).

### Inequalities of the form \(\cot x \gt a,\) \(\cot x \ge a,\) \(\cot x \lt a,\) \(\cot x \le a\)

Figure 7.

Figure 8.

#### Inequality \(\cot x \gt a\)

- For any value of \(a\), the solution of the inequality \(\cot x \gt a\) has the form

\(\pi n \lt x \lt \text {arccot } a + \pi n,\) \(n \in \mathbb{Z}\) (Fig.7).

#### Inequality \(\cot x \ge a\)

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

\(\pi n \lt x \le \text {arccot } a + \pi n,\) \(n \in \mathbb{Z}\) (Fig.7).

#### Inequality \(\cot x \lt a\)

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

\(\text {arccot } a + \pi n \lt x\) \(\lt \pi + \pi n,\) \(n \in \mathbb{Z}\) (Fig.8).

#### 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

\(\text {arccot } a + \pi n \le x\) \(\lt \pi + \pi n,\) \(n \in \mathbb{Z}\) (Fig.8).

### Related Pages

- Definition and Graphs of Trigonometric Functions
- Basic Trigonometric Identities
- Cofunction and Reduction Identities
- Relationships between Trigonometric Functions
- Addition and Subtraction Formulas
- Double and Multiple Angle Formulas
- Half-Angle Formulas
- Sum-to-Product Identities
- Product-to-Sum Identities
- Derivatives of Trigonometric Functions
- The Indefinite Integral and Basic Formulas of Integration. Table of Integrals.
- Basic Trigonometric Equations
- Algebraic Inequalities