Precalculus

Trigonometry

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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

\[\arcsin a + 2\pi n \lt x \lt \pi - \arcsin a + 2\pi n, \;n \in \mathbb{Z}.\]
Solution of the inequality involving the sine function (case 1)
Figure 1.

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

\[-\pi - \arcsin a + 2\pi n \lt x \lt \arcsin a + 2\pi n,\;n \in \mathbb{Z}.\]
Solution of the inequality involving the sine function (case 2)
Figure 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 \(a \lt -1\), 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

\[-\arccos a + 2\pi n \lt x \lt \arccos a + 2\pi n,\;n \in \mathbb{Z}.\]
Solution of the inequality involving cosine (case 1)
Figure 3.

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

\[\arccos a + 2\pi n \lt x \lt 2\pi - \arccos a + 2\pi n,\;n \in \mathbb{Z}.\]
Solution of the inequality involving cosine (case 2)
Figure 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: \(x \in \varnothing.\)

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

\[\arctan a + \pi n \lt x \lt \pi/2 + \pi n,\;n \in \mathbb{Z}.\]
Solution of the inequality involving tangent (case 1)
Figure 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

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

Inequality \(\tan x \lt a\)

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

\[-\pi/2 + \pi n \lt x \lt \arctan a + \pi n,\;n \in \mathbb{Z}.\]
Solution of the inequality involving tangent (case 2)
Figure 6.

Inequality \(\tan x \le a\)

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

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

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

\[\pi n \lt x \lt \text {arccot } a + \pi n,\;n \in \mathbb{Z}.\]
Solution of the inequality involving cotangent (case 1)
Figure 7.

Inequality \(\cot x \ge a\)

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

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

Inequality \(\cot x \lt a\)

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

\[\text {arccot } a + \pi n \lt x \lt \pi + \pi n,\;n \in \mathbb{Z}.\]
Solution of the inequality involving cotangent (case 2)
Figure 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

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