Formulas

Trig Identities

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Basic Trigonometric Equations

    • Angles (arguments of functions): \(x,\) \({x_1},\) \({x_2}\)
    Set of integers: \(\mathbb{Z}\)
    Integer: \(n\)
    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 equation involving trigonometric functions of an unknown angle is called a trigonometric equation.
    2. Basic trigonometric equations have the form
      \(\sin x = a,\) \(\cos x = a,\) \(\tan x = a,\) \(\cot x = a,\)
      where \(x\) is an unknown, \(a\) is any real number.

      3. Equation \(\sin x = a\)

    3. If \(\left| a \right| \gt 1\), the equation \(\sin x = a\) has no solutions.
    4. If \(\left| a \right| \le 1,\) the general solution of the equation \(\sin x = a\) is written as
      \(x = {\left( { – 1} \right)^n}\arcsin a + \pi n,\) \(n \in \mathbb{Z}.\)
      This formula contains two branches of solutions:
      \({x_1} = \arcsin a + 2\pi n\),  \({x_2} = \pi – \arcsin a + 2\pi n,\) \(n \in \mathbb{Z}\).
      5. Solution of the equation sin(x)=a
    5. In the simple case \(\sin x = 1\) the solution has the form
      \(x = \pi/2 + 2\pi n,\) \(n \in \mathbb{Z}\).
    6. Similarly, the solution of the equation \(\sin x = -1\) is given by
      \(x = -\pi/2 + 2\pi n,\) \(n \in \mathbb{Z}\).
    7. Case \(\sin x = 0\) (zeroes of the sine)
      \(x = \pi n,\) \(n \in \mathbb{Z}\).

      8. Equation \(\cos x = a\)

    8. If \(\left| a \right| \gt 1,\) the equation \(\cos x = a\) has no solutions.
    9. If \(\left| a \right| \le 1,\) the general solution of the equation \(\cos x = a\) has the form
      \(x = \pm \arccos a + 2\pi n,\) \(n \in \mathbb{Z}.\)
      This formula includes two sets of solutions:
      \({x_1} = \arccos a + 2\pi n\), \({x_2} = -\arccos a + 2\pi n,\) \(n \in \mathbb{Z}\).
      10. Solution of the equation cos(x)=a
    10. In the case \(\cos x = 1\), the solution is written as
      \(x = 2\pi n,\) \(n \in \mathbb{Z}\).
    11. Case \(\cos x = -1\)
      \(x = \pi + 2\pi n,\) \(n \in \mathbb{Z}\).
    12. Case \(\cos x = 0\) (zeroes of the cosine)
      \(x = \pi/2 + \pi n,\) \(n \in \mathbb{Z}\).

      13. Equation \(\tan x = a\)

    13. For any value of \(a\), the general solution of the equation \(\tan x = a\) has the form
      \(x = \arctan a + \pi n,\) \(n \in \mathbb{Z}.\)
      14. Solution of the equation tan(x)=a
    14. Case \(\tan x = 0\) (zeroes of the tangent)
      \(x = \pi n,\) \(n \in \mathbb{Z}.\)

      15. Equation \(\cot x = 0\)

    15. For any value of\(a\), the general solution of the trigonometric equation \(\cot x = 0\) is written as
      \(x = \text {arccot } a + \pi n,\) \(n \in \mathbb{Z}.\)
      16. Solution of the equation cot(x)=a
    16. Case \(\cot x = 0\) (zeroes of the cotangent)
      \(x = \pi/2 + \pi n,\) \(n \in \mathbb{Z}.\)