Formulas and Tables

Trigonometry

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: \(\arcsin a,\) \(\arccos a,\) \(\arctan a,\) \(\text {arccot } a\)

  1. An inequality involving trigonometric functions of an unknown angle is called a trigonometric inequality.
  2. 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 \(\sin x \gt a,\) \(\sin x \ge a,\) \(\sin x \lt a,\) \(\sin x \le a\)

Solution of the inequality involving the sine function (case 1)

Figure 1.

Solution of the inequality involving the sine function (case 2)

Figure 2.

Inequality \(\sin x \gt a\)

  1. If \(\left| a \right| \ge 1\), the inequality \(\sin x \gt a\) has no solutions: \(x \in \emptyset\)
  2. If \(a \lt -1\), the solution of the inequality \(\sin x \gt a\) is any real number: \(x \in \mathbb{R}\)
  3. 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 \(\sin x \ge a\)

  1. If \(a \gt 1\), the inequality \(\sin x \ge a\) has no solutions: \(x \in \emptyset\)
  2. If \(a \le -1\), the solution of the inequality \(\sin x \ge a\) is any real number: \(x \in \mathbb{R}\)
  3. Case \(a = 1\)
    \(x = \pi/2 +2\pi n,\) \(n \in \mathbb{Z}\)
  4. 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 \(\sin x \lt a\)

  1. If \(a \gt 1\), the solution of the inequality \(\sin x \lt a\) is any real number: \(x \in \mathbb{R}\)
  2. If \(a \le -1\), the inequality \(\sin x \lt a\) has no solutions: \(x \in \emptyset\)
  3. 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 \(\sin x \le a\)

  1. If \(a \ge 1\), the solution of the inequality \(\sin x \le a\) is any real number: \(x \in \mathbb{R}\)
  2. If \(a < -1\), the inequality \(\sin x \le a\) has no solutions: \(x \in \emptyset\)
  3. Case \(a = -1\)
    \(x = -\pi/2 + 2\pi n,\) \(n \in \mathbb{Z}\)
  4. 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 \(\cos x \gt a,\) \(\cos x \ge a,\) \(\cos x \lt a,\) \(\cos x \le a\)

Solution of the inequality involving cosine (case 1)

Figure 3.

Solution of the inequality involving cosine (case 2)

Figure 4.

Inequality \(\cos x \gt a\)

  1. If \(a \ge 1\), the inequality \(\cos x \gt a\) has no solutions: \(x \in \emptyset\)
  2. If \(a \lt -1\), the solution of the inequality \(\cos x \gt a\) is any real number: \(x \in \mathbb{R}\)
  3. 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 \(\cos x \ge a\)

  1. If \(a \gt 1\), the inequality \(\cos x \ge a\) has no solutions: \(x \in \emptyset\)
  2. If \(a \le -1\), the solution of the inequality \(\cos x \ge a\) is any real number: \(x \in \mathbb{R}\)
  3. Case \(a = 1\)
    \(x = 2\pi n,\) \(n \in \mathbb{Z}\)
  4. 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 \(\cos x \lt a\)

  1. If \(a \gt 1\), the inequality \(\cos x \lt a\) is true for any real value of \(x\): \(x \in \mathbb{R}\)
  2. If \(a \le -1\), the inequality \(\cos x \lt a\) has no solutions: \(x \in \emptyset\)
  3. 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 \(\cos x \le a\)

  1. If \(a \ge 1\), the solution of the inequality \(\cos x \le a\) is any real number: \(x \in \mathbb{R}\)
  2. If \(a \lt -1\), the inequality \(\cos x \le a\) has no solutions: \(x \in \emptyset\)
  3. Case \(a = -1\)
    \(x = \pi + 2\pi n,\) \(n \in \mathbb{Z}\)
  4. 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 \(\tan x \gt a,\) \(\tan x \ge a,\) \(\tan x \lt a,\) \(\tan x \le a\)

Solution of the inequality involving tangent (case 1)

Figure 5.

Solution of the inequality involving tangent (case 2)

Figure 6.

Inequality \(\tan x \gt a\)

  1. 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 \(\tan x \ge a\)

  1. 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 \(\tan x \lt a\)

  1. 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 \(\tan x \le a\)

  1. 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 \(\cot x \gt a,\) \(\cot x \ge a,\) \(\cot x \lt a,\) \(\cot x \le a\)

Solution of the inequality involving cotangent (case 1)

Figure 7.

Solution of the inequality involving cotangent (case 2)

Figure 8.

Inequality \(\cot x \gt a\)

  1. 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 \(\cot x \ge a\)

  1. 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 \(\cot x \lt a\)

  1. 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 \(\cot x \le a\)

  1. 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).