Precalculus

Trigonometry

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Basic Trigonometric Inequalities

Unknown variable (angle): x
Set of integers: ℤ
Integer: n
Set of real numbers: ℝ
Real number: a

Trigonometric functions: sin x, cos x, tan x, cot x
Inverse trigonometric functions: arcsin a, arccos a, arctan a, 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 \(\sin x \gt a,\) \(\sin x \ge a,\) \(\sin x \lt a,\) \(\sin x \le a\)

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

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

Solved Problems

Example 1.

Solve the inequality

\[\sin x \ge -\frac{1}{2}.\]

Solution.

Mark on the \(y-\)axis a point \(y = -\frac{1}{2}.\) Find the angles corresponding to the given value of the sine:

\[\sin x = -\frac{1}{2}, \Rightarrow\]
\[x_1 = \arcsin{\left({-\frac{1}{2}}\right)} = - \arcsin{\left({\frac{1}{2}}\right)} = -\frac{\pi}{6};\]
\[x_2 = \pi - {x_1} = \pi - \left({-\frac{\pi}{6}}\right) = \pi + \frac{\pi}{6} = \frac{7\pi}{6}.\]
Solution of the basic inequality sin x >= -1/2
Figure 9.

The solution of inequality \(\sin x \ge -\frac{1}{2}\) on the unit circle is represented as the sector \(\left[-\frac{\pi}{6}, \frac{7\pi}{6}\right].\) Do not forget to add periodic terms in the answer:

\[-\frac{\pi}{6} + 2\pi n \le x \le \frac{7\pi}{6} + 2\pi n,\;n \in \mathbb{Z}.\]

Example 2.

Solve the inequality

\[\cos x \lt -\frac{\sqrt{2}}{2}.\]

Solution.

Solving the equation \(\cos x = -\frac{\sqrt{2}}{2}\) we find:

\[x = \pm\arccos \left({-\frac{\sqrt{2}}{2}}\right) + 2\pi n = \pm\left({\pi - \arccos{\frac{\sqrt{2}}{2}}}\right) + 2\pi n = \pm\left({\pi - \frac{\pi}{4}}\right) + 2\pi n = \pm\frac{3\pi}{4} + 2\pi n,\;n \in \mathbb{Z}.\]

Notice that the angle \(-\frac{3\pi}{4}\) can be written as \(\frac{5\pi}{4}.\) Hence, the solution of the inequality lies within the interval \(\left[{\frac{3\pi}{4}, \frac{5\pi}{4}}\right].\)

Solution of the basic inequality cos x < -sqrt(2)/2
Figure 10.

The answer is given by

\[\frac{3\pi}{4} + 2\pi n \lt x \lt \frac{5\pi}{4} + 2\pi n,\;n \in \mathbb{Z}.\]

Example 3.

Solve the inequality

\[\tan x \gt -\sqrt{3}.\]

Solution.

The solution of the inequality \(\tan x \gt a\) is written in the form

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

In our case:

\[\arctan a = \arctan\left({-\sqrt{3}}\right) = - \arctan\sqrt{3} = -\frac{\pi}{3}.\]

Therefore, we have:

\[-\frac{\pi}{3} + \pi n \lt x \lt \pi/2 + \pi n,\;n \in \mathbb{Z}.\]
Solution of the inequality tan x > -sqrt(3)
Figure 11.

Example 4.

Solve the inequality

\[\cot x \ge \frac{1}{\sqrt{3}}.\]

Solution.

We know that the solution of the non-strict inequality \(\cot x \ge a\) is given by

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

Since

\[\text{arccot}\frac{1}{\sqrt{3}} = \frac{\pi}{3},\]

we can write

\[\pi n \lt x \le \frac{\pi}{3} + \pi n,\;n \in \mathbb{Z}.\]
Solution of the inequality cot x >= 1/sqrt(3)
Figure 12.