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Systematic Approach to Trigonometric Equations

May 14, 2025

Solving Trigonometric Equations

Introduction

  • Trigonometric equations involve sine, cosine, and tangent.
  • Structure to follow:
    1. Simplify the equation.
    2. Factorize if possible.
    3. Obtain the reference angle.
    4. Choose the correct quadrants.
    5. Solve the equation.

Types of Solutions

  • General Solution: No interval is given.
  • Specific Solution: Interval is provided.
    • Both methods are identical until the final step for specific solutions.

Example Solving Process

  1. Equation: ( \sin x = 0.4 )

    • Already simplified and factorized.
    • Reference Angle: Use calculator (shift+sin), result: 23.58°.
    • Quadrants: Sine is positive in Quadrants 1 & 2.
    • Solve: Quadrant 1:
      • ( x = 23.58 + k \times 360 )
    • Solve: Quadrant 2:
      • ( x = 156.42 + k \times 360 )
    • Specific Solutions: Check against interval 0° to 360°, resulting in x = 23.58° and x = 156.42°.

  2. Equation: ( \sin (x-20) = -0.3 )

    • Already simplified and factorized.
    • Reference Angle: Ignore negative, use ( \sin^{-1}(0.3) = 17.46° ).
    • Quadrants: Sine is negative in Quadrants 3 & 4.
    • Solve: Quadrant 3:
      • ( x - 20 = 180 + 17.46 + k \times 360 )
      • ( x = 217.46 + k \times 360 )
    • Solve: Quadrant 4:
      • ( x - 20 = 360 - 17.46 + k \times 360 )
      • ( x = 362.54 + k \times 360 )
    • General Solution: No interval, so solutions are left as is.

  3. Equation: ( \cos(2x-10) = 0.5 )

    • Simplify by dividing by 2.
    • Reference Angle: ( \cos^{-1}(0.5) = 60° ).
    • Quadrants: Cosine is positive in Quadrants 1 & 4.
    • Solve: Quadrant 1:
      • ( 2x - 10 = 60 + k \times 360 )
      • ( x = 35 + k \times 180 )
    • Solve: Quadrant 4:
      • ( 2x - 10 = 360 - 60 + k \times 360 )
      • ( x = 155 + k \times 180 )
    • Specific Solutions: Check k values within interval -360° to 360°, results in eight solutions.

Tips

  • Always ensure calculator is in degree mode, not radians.
  • General solutions include ( k \times 360 ), specific solutions require checking within intervals.

Conclusion: Follow the systematic approach to solve trigonometric equations effectively, ensuring proper handling of angles, quadrants, and solutions within given intervals.

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