General solution of standard trigonometrical equations

Introduction: The general solution of a trigonometric equation refers to an expression that represents all possible solutions by incorporating an integer parameter. This concept is particularly important for equations involving periodic functions such as sine, cosine, tangent, cotangent, secant, and cosecant.

1. Periodicity of Trigonometric Functions:

  • Principal Periods:
    • Sine and cosine have a principal period of 2π, while tangent, cotangent, secant, and cosecant have a principal period of π.
    • This means the functions repeat their values every 2π or π radians.

2. General Solution Formulas:

  • Linear Trigonometric Equations:

    • sin(θ)=aor cos(θ)=a has the general solution θ=sin1(a)+2nπ or θ=cos1(a)+2nπ, where n is an integer.
  • Quadratic Trigonometric Equations:

    • sin2(θ)=a or cos2(θ)=a has the general solution θ=sin1(a)+2nπ or θ=cos1(a)+2nπ, where n is an integer.
  • Tangent and Cotangent Equations:

    • tan(θ)=a or cot(θ)=a has the general solution θ=tan1(a)+nπ or θ=cot1(a)+nπ, where n is an integer.
  • Secant and Cosecant Equations:

    • sec(θ)=a or csc(θ)=a has the general solution θ=sec1(a)+nπ or θ=csc1(a)+nπ, where n is an integer.

3. Periodic Adjustments:

  • Adjustments for Periodicity:
    • Due to the periodic nature of trigonometric functions, solutions may repeat at intervals of 2π or π. Adjust the general solution accordingly by adding 2nπ or nπ to represent all possible solutions.

4. Verification:

  • Check Solutions:
    • Verify obtained solutions by substituting them back into the original equation. Ensure they satisfy the equation within the given range.

5. Examples:

  • Linear Equation:

    • Solve sin(θ)=12. The general solution is θ=sin1(12)+2nπ.
  • Quadratic Equation:

    • Solve cos2(θ)=34. The general solution is θ=cos1(32)+2nπ.
  • Tangent Equation:

    • Solve tan(θ)=1. The general solution is θ=tan1(1)+nπ.