General Solution of Specific Trigonometric Equations

Introduction: The general solution of trigonometric equations provides a comprehensive expression for all possible solutions, considering the periodic nature of trigonometric functions. This is particularly relevant for equations involving inverse trigonometric functions, multiple angles, and special values.

1. Inverse Trigonometric Equations:

  • Example:

    • Consider the equation sin1(x)=π4.
  • General Solution:

    • The general solution for sin1(x)=π4 is x=sin(π4)+2nπ, where n is an integer.
    • This accounts for the periodic nature of the sine function.

2. Trigonometric Equations with Multiple Angles:

  • Example:

    • Solve cos(2x)+cos(x)=1.
  • General Solution:

    • The general solution for cos(2x)+cos(x)=1 involves expressing x in terms of a multiple angle: x=π3+nπ

3. Trigonometric Equations with Special Values:

  • Example:

    • Solve tan(x)=1.
  • General Solution:

    • The general solution for tan(x)=1 is x=tan1(1)+nπ.
    • This takes into account the periodicity of the tangent function.

4. Periodicity Consideration:

  • Principal Periods:

    • Be aware of the principal periods of trigonometric functions. Sine and cosine have a principal period of 2π, while tangent has a principal period of π.
  • Adjustments for Periodicity:

    • Make adjustments to the general solution by adding 2nπor nπ to represent all possible solutions.

5. Verification:

  • Check Solutions:
    • Always verify obtained solutions by substituting them back into the original equation to ensure consistency.

6. Examples:

  • Inverse Trigonometric Equation:

    • Solve cos1(x)=π3. The general solution is x=cos(π3)+2nπ.
  • Equation with Multiple Angles:

    • Solve sin(2x)=cos(x). The general solution is x=π4+nπ.
  • Equation with Special Values:

    • Solve sin(x)=22. The general solution is x=sin1(22)+2nπ.