Formulas

Trig Identities

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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: \(\arcsin a,\) \(\arccos a,\) \(\arctan a,\) \(\text {arccot } a\)
    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 \(\sin x \gt a,\) \(\sin x \ge a,\) \(\sin x \lt a,\) \(\sin x \le a\)

      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 \(\sin x \gt a\)

    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 \(\cos x \gt a,\) \(\cos x \ge a,\) \(\cos x \lt a,\) \(\cos x \le a\)

      Solution of the inequality involving cosine (case 1)
      Figure 3.
      Solution of the inequality involving cosine (case 2)
      Figure 4.

      Inequality \(\cos x \gt a\)

    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 \(\tan x \gt a,\) \(\tan x \ge a,\) \(\tan x \lt a,\) \(\tan x \le a\)

      Solution of the inequality involving tangent (case 1)
      Figure 5.
      Solution of the inequality involving tangent (case 2)
      Figure 6.

      Inequality \(\tan x \gt a\)

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

      Solution of the inequality involving cotangent (case 1)
      Figure 7.
      Solution of the inequality involving cotangent (case 2)
      Figure 8.

      Inequality \(\cot x \gt a\)

    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 \(8\)).