MATH04 Pre Calculus Study Notes

Course Outcome 4

Lesson 2: Trigonometric Equations

Introduction to Trigonometric Equations

• Definition: Equations that involve trigonometric functions are called trigonometric equations.

• Key Concept: Solving a trigonometric equation is similar to solving an algebraic equation, involving the identification and finding of values of the variable that satisfy the equation.

• Techniques for Solving:

  • Isolating the variable

  • Collecting like terms

  • Factoring

  • Substituting trigonometric identities

Types of Trigonometric Equations

Identical Trigonometric Equation

• Definition: A trigonometric equation that is true for all permissible values of the unknown variable for which the equation is defined.

• Example:

  • ext{sin}^2 x + ext{cos}^2 x = 1

Conditional Trigonometric Equation

• Definition: A trigonometric equation that is true for some, but not all, permissible values of the unknown variable.

• Examples:

  1. 2 ext{sin} x - 1 = 0

  2. ext{tan}^2 2x - 1 = 0

Guidelines in Solving Trigonometric Equations

1. Single Function of a Single Angle: Use algebraic techniques to solve for the angle.

2. Quadratic Equation: If it contains a single function of the same angle, factor it; otherwise, use the quadratic formula.

3. Multiple Functions of the Same Angle: Substitute trigonometric identities to obtain a single function.

4. Multiple Angles: Substitute trigonometric identities to obtain a function of a single angle.

Exercise: Solve Each Equation for Exact Solutions in the Interval 0 ≤ x < 360°

a) 2 ext{sin}^2 x = 1

b) ext{sin}^2 x + 3 ext{sin} x + 2 = 0

c) 2 ext{cos}^2 x + 1 = -3 ext{cos} x

d) 4 ext{cos}^3 x = 3 ext{cos} x

e) 3 ext{tan}^2 x + 5 ext{tan} x - 1 = 0

Expanded Table for Special and Quadrantal Angles

• Angle (Radians) | Angle (Degrees) | sin | cos | tan | csc | sec | cot

  • 0 | 0° | 0 | 1 | 0 | Undef | 1 | Undef

  • rac{ ext{π}}{6} | 30° | rac{1}{2} | rac{ ext{√3}}{2} | rac{ ext{√3}}{3} | 2 | 2 rac{ ext{√3}}{3} | ext{√3}

  • rac{ ext{π}}{4} | 45° | rac{ ext{√2}}{2} | rac{ ext{√2}}{2} | 1 | ext{√2} | ext{√2} | 1

  • rac{ ext{π}}{3} | 60° | rac{ ext{√3}}{2} | rac{1}{2} | rac{ ext{√3}}{3} | 2 | rac{2 ext{√3}}{3} | 2

  • rac{ ext{π}}{2} | 90° | 1 | 0 | Undef | 1 | Undef | 0

  • rac{2 ext{π}}{3} | 120° | rac{ ext{√3}}{2} | - rac{1}{2} | - rac{ ext{√3}}{3} | -2 | - rac{2 ext{√3}}{3} | - ext{√3}

  • rac{3 ext{π}}{4} | 135° | rac{ ext{√2}}{2} | - rac{ ext{√2}}{2} | -1 | ext{√2} | - ext{√2} | -1

  • rac{5 ext{π}}{6} | 150° | rac{1}{2} | - rac{ ext{√3}}{2} | - rac{ ext{√3}}{3} | -2 | - rac{2 ext{√3}}{3} | - ext{√3}

  • ext{π} | 180° | 0 | -1 | 0 | -1 | 0 | -1

- … (continues for other angles)

Solutions to Exercises

a) 2 ext{sin}^2 x = 1

• Step 1: Divide both sides by 2: ext{sin}^2 x = rac{1}{2}

• Step 2: Taking square roots: ext{sin} x = rac{ ext{√2}}{2} ext{ or } - rac{ ext{√2}}{2}

• Step 3: Solutions: x = 45°, 135°, 225°, 315°

b) ext{sin}^2 x + 3 ext{sin} x + 2 = 0

• Factoring: ( ext{sin}x + 2)( ext{sin}x + 1) = 0

• Solve:

  1. ext{sin} x = -2 - No solution as sine cannot exceed these bounds.

  2. ext{sin} x = -1

• Solutions: x = 270°

c) 2 ext{cos}^2 x + 1 = -3 ext{cos} x

• Substituting x= ext{cos} x:

  1. 2x^2 + 3x + 1 = 0

• Factoring: (2x + 1)(x + 1) = 0

  1. x = - rac{1}{2}

  2. x = -1

• Find angles: ext{cos} x = - rac{1}{2} ext{ or } -1

• Results: Solutions: x = 120°, 240°, 180°

d) 4 ext{cos}^3 x - 3 ext{cos} x = 0

• Factoring the equation:

  • ext{cos} x(4 ext{cos}^2 x - 3) = 0 \

• Solutions for ext{cos}x = 0: x = 90°, 270°

• For 4 ext{cos}^2 x - 3 = 0

  • ext{cos}^2 x = rac{3}{4}

  • ext{cos} x = rac{ ext{√3}}{2}, - rac{ ext{√3}}{2}

  • Resulting angles: x = 30°, 150°, 210°, 330°

e) 3 ext{tan}^2 x + 5 ext{tan} x - 1 = 0

• Quadratic Formula (Letting x = ext{tan} x in the quadratic):

  • an x = rac{-b ext{±} ext{√}(b^2 - 4ac)}{2a}

  • Solutions: Use calculator: an x = 0.1805 ext{ and } -1.8471

• Final Angles: x = 10.23°, 190.23°, 298.43°, -61.57° ext{ rounded into quadrant.}

Additional Exercises

1. ext{cos} 3x + ext{cos} 5x + ext{cos} 2x + ext{sin} 5x ext{sin} 2x - 1 = 0

2. ext{sin} x + ext{cos} x = ext{√2}

3. 3 ext{cos}^2 x + 3 ext{sin}^2 x = 1

4. 3 ext{sin}^3 x = 0

Further Exercises

1. Solutions in the interval [0, 2 ext{π}):

   • a) 2 ext{sin} x + ext{√3} = 0

   • b) ext{cos} x = -0.4298$$

Conclusion

• Understanding and solving trigonometric equations is essential for further studies in trigonometry and calculus.

• Utilizing trigonometric identities and algebraic techniques expands problem-solving capabilities in mathematics.