Lecture Notes: Solving Trigonometric Equations

Key Concepts

• Trigonometric Equations: Equations involving trigonometric functions like sine, cosine, tangent, etc.
• Reciprocal Functions: For example, cosecant (csc) x is 1/sin x.
• Solving Techniques: Transforming equations, factoring, using identities, and understanding reference angles and unit circles.

Example Problems

Problem 1: Solving 2 sin x + csc x = 0

• Step 1: Recognize csc x as 1/sin x. The equation is:
  • [2 \sin x + \frac{1}{\sin x} = 0]
• Step 2: Multiply every term by sin x to eliminate the fraction:
  • [2 \sin^2 x = -1]
• Conclusion: After simplifying, (\sin^2 x = -1/2), which is impossible (squared terms cannot be negative). Thus, the equation has no solution.

Problem 2: Solving cos x - sin x = 0

• Step 1: Rearrange to cos x = sin x.
• Step 2: Divide by cos x to get tan x = 1.
• Solution:
  • Use a calculator or unit circle: tan(45°)=1.
  • Solutions in degrees: 45° (Quadrant 1), 225° (Quadrant 3),
  • In radians: π/4 and 5π/4.
• General Solution:
  • Tangent is periodic every π, so: π/4 + πn.

Problem 3: Solving 3 tan³ x = tan x

• Step 1: Rearrange to 3 tan³ x - tan x = 0.
• Step 2: Factor out tan x: (tan x (3 tan^2 x - 1) = 0).
• Solutions:
  • (tan x = 0) at x = 0, π.
  • Solving (3 tan^2 x = 1), gives (tan x = ± \sqrt{3}/3).
• Reference Angles:
  • 30° in 30-60-90 triangle gives solutions:
  • Quadrant 1: 30°, Quadrant 2: 150°, Quadrant 3: 210°, Quadrant 4: 330°.
  • In radians: π/6, 5π/6, 7π/6, 11π/6.
• General Solutions:
  • For 0 and π: Add πn.
  • Use periodicity for others.

Problem 4: Solving 2 sin² x + 3 sin x + 1 = 0

• Step 1: Recognize as quadratic: Let a = sin x, then solve (2a^2 + 3a + 1 = 0).
• Step 2: Factor: (2a+1)(a+1)=0.
• Solution:
  • Set each factor to zero: sin x = -1, sin x = -1/2.
  • Solutions are: x = 3π/2 (270°), 210°, 330°.
  • Converting to radians: 7π/6, 11π/6.
• General Solutions:
  • Each angle can be expressed with 2πn for unrestricted domain.

Additional Tips

• Triangles: Know 30-60-90 and 45-45-90 triangles for solving trig equations.
• Unit Circle: Familiarize with key points for angles in radians and degrees.
• Periodicity: Use properties of periodic functions to find general solutions.