# 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.
3. ### 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 $$\sin x \gt a$$

4. If $$\left| a \right| \ge 1$$, the inequality $$\sin x \gt a$$ has no solutions: $$x \in \emptyset$$
5. If $$a \lt -1$$, the solution of the inequality $$\sin x \gt a$$ is any real number: $$x \in \mathbb{R}$$
6. For $$-1 \le a \lt 1$$, the solution of the inequality $$\sin x \gt a$$ is expressed in the form
$$\arcsin a + 2\pi n \lt x$$ $$\lt \pi – \arcsin a + 2\pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$1$$).
7. #### Inequality $$\sin x \ge a$$

8. If $$a \gt 1$$, the inequality $$\sin x \ge a$$ has no solutions: $$x \in \emptyset$$
9. If $$a \le -1$$, the solution of the inequality $$\sin x \ge a$$ is any real number: $$x \in \mathbb{R}$$
10. Case $$a = 1$$
$$x = \pi/2 +2\pi n,$$ $$n \in \mathbb{Z}$$
11. 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
$$\arcsin a + 2\pi n \le x$$ $$\le \pi – \arcsin a + 2\pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$1$$).
12. #### Inequality $$\sin x \lt a$$

13. If $$a \gt 1$$, the solution of the inequality $$\sin x \lt a$$ is any real number: $$x \in \mathbb{R}$$
14. If $$a \le -1$$, the inequality $$\sin x \lt a$$ has no solutions: $$x \in \emptyset$$
15. For $$-1 \lt a \le 1$$, the solution of the inequality $$\sin x \lt a$$ lies in the interval
$$-\pi – \arcsin a + 2\pi n \lt x$$ $$\lt \arcsin a + 2\pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$2$$).
16. #### Inequality $$\sin x \le a$$

17. If $$a \ge 1$$, the solution of the inequality $$\sin x \le a$$ is any real number: $$x \in \mathbb{R}$$
18. If $$a < -1$$, the inequality $$\sin x \le a$$ has no solutions: $$x \in \emptyset$$
19. Case $$a = -1$$
$$x = -\pi/2 + 2\pi n,$$ $$n \in \mathbb{Z}$$
20. For $$-1 \lt a \lt 1$$, the solution of the non-strict inequality $$\sin x \le a$$ is in the interval
$$-\pi – \arcsin a + 2\pi n \le x$$ $$\le \arcsin a + 2\pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$2$$).
21. ### 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 $$\cos x \gt a$$

22. If $$a \ge 1$$, the inequality $$\cos x \gt a$$ has no solutions: $$x \in \emptyset$$
23. If $$a \lt -1$$, the solution of the inequality $$\cos x \gt a$$ is any real number: $$x \in \mathbb{R}$$
24. For $$-1 \le a \lt 1$$, the solution of the inequality $$\cos x \gt a$$ has the form
$$-\arccos a + 2\pi n \lt x$$ $$\lt \arccos a + 2\pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$3$$).
25. #### Inequality $$\cos x \ge a$$

26. If $$a \gt 1$$, the inequality $$\cos x \ge a$$ has no solutions: $$x \in \emptyset$$
27. If $$a \le -1$$, the solution of the inequality $$\cos x \ge a$$ is any real number: $$x \in \mathbb{R}$$
28. Case $$a = 1$$
$$x = 2\pi n,$$ $$n \in \mathbb{Z}$$
29. For $$-1 \lt a \lt 1$$, the solution of the non-strict inequality $$\cos x \ge a$$ is expressed by the formula
$$-\arccos a + 2\pi n \le x$$ $$\le \arccos a + 2\pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$3$$).
30. #### Inequality $$\cos x \lt a$$

31. If $$a \gt 1$$, the inequality $$\cos x \lt a$$ is true for any real value of $$x$$: $$x \in \mathbb{R}$$
32. If $$a \le -1$$, the inequality $$\cos x \lt a$$ has no solutions: $$x \in \emptyset$$
33. For $$-1 < a \le 1$$, the solution of the inequality $$\cos x \lt a$$ is written in the form $$\arccos a + 2\pi n \lt x$$ $$\lt 2\pi – \arccos a + 2\pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$4$$).
34. #### Inequality $$\cos x \le a$$

35. If $$a \ge 1$$, the solution of the inequality $$\cos x \le a$$ is any real number: $$x \in \mathbb{R}$$
36. If $$a \lt -1$$, the inequality $$\cos x \le a$$ has no solutions: $$x \in \emptyset$$
37. Case $$a = -1$$
$$x = \pi + 2\pi n,$$ $$n \in \mathbb{Z}$$
38. For $$-1 \lt a \lt 1$$, the solution of the non-strict inequality $$\cos x \le a$$ is written as
$$\arccos a + 2\pi n \le x$$ $$\le 2\pi – \arccos a + 2\pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$4$$).
39. ### 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 $$\tan x \gt a$$

40. For any real value of $$a$$, the solution of the strict inequality $$\tan x \gt a$$ has the form
$$\arctan a + \pi n \lt x$$ $$\lt \pi/2 + \pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$5$$).
41. #### Inequality $$\tan x \ge a$$

42. For any real value of $$a$$, the solution of the inequality $$\tan x \ge a$$ is expressed in the form
$$\arctan a + \pi n \le x$$ $$\lt \pi/2 + \pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$5$$).
43. #### Inequality $$\tan x \lt a$$

44. For any value of $$a$$, the solution of the inequality $$\tan x \lt a$$ is written in the form
$$-\pi/2 + \pi n \lt x$$ $$\lt \arctan a + \pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$6$$).
45. #### Inequality $$\tan x \le a$$

46. For any value of $$a$$, the inequality $$\tan x \le a$$ has the following solution:
$$-\pi/2 + \pi n \lt x$$ $$\le \arctan a + \pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$6$$).
47. ### 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 $$\cot x \gt a$$

48. For any value of $$a$$, the solution of the inequality $$\cot x \gt a$$ has the form
$$\pi n \lt x \lt \text {arccot } a + \pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$7$$).
49. #### Inequality $$\cot x \ge a$$

50. The non-strict inequality $$\cot x \ge a$$ has the similar solution:
$$\pi n \lt x \le \text {arccot } a + \pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$7$$).
51. #### Inequality $$\cot x \lt a$$

52. For any value of $$a$$, the solution of the inequality $$\cot x \lt a$$ lies on the open interval
$$\text {arccot } a + \pi n \lt x$$ $$\lt \pi + \pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$8$$).
53. #### Inequality $$\cot x \le a$$

54. For any value of $$a$$, the solution of the non-strict inequality $$\cot x \le a$$ is in the half-open interval
$$\text {arccot } a + \pi n \le x$$ $$\lt \pi + \pi n,$$ $$n \in \mathbb{Z}$$ (Figure $$8$$).