Trigonometric Ratios and Equations

 

Trigonometric Ratios and Equations: A Quick Reference

Trigonometric Ratios

• Sine (sin): Opposite side / Hypotenuse  
• Cosine (cos): Adjacent side / Hypotenuse  
• Tangent (tan): Opposite side / Adjacent side
• Cosecant (csc): 1 / sin  
• Secant (sec): 1 / cos  
• Cotangent (cot): 1 / tan

Pythagorean Identity

• sin²θ + cos²θ = 1

Trigonometric Identities

• Double Angle Formulas:
  • sin(2θ) = 2sinθcosθ
  • cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
  • tan(2θ) = 2tanθ / (1 - tan²θ)
• Half Angle Formulas:
  • sin(θ/2) = ±√[(1 - cosθ) / 2]
  • cos(θ/2) = ±√[(1 + cosθ) / 2]
  • tan(θ/2) = sinθ / (1 + cosθ) = (1 - cosθ) / sinθ
• Sum and Difference Formulas:
  • sin(α + β) = sinαcosβ + cosαsinβ
  • sin(α - β) = sinαcosβ - cosαsinβ
  • cos(α + β) = cosαcosβ - sinαsinβ
  • cos(α - β) = cosαcosβ + sinαsinβ
  • tan(α + β) = (tanα + tanβ) / (1 - tanαtanβ)  
  • tan(α - β) = (tanα - tanβ) / (1 + tanαtanβ)  

Trigonometric Equations

• Solving trigonometric equations often involves using trigonometric identities and algebraic techniques.  
• Common methods include:
  • Factoring
  • Using quadratic formula  
  • Using trigonometric identities
  • Graphing

Example: Solve sin(2x) = cos(x)

• Use the double angle formula for sin(2x): 2sin(x)cos(x) = cos(x)
• Divide both sides by cos(x): 2sin(x) = 1
• Solve for sin(x): sin(x) = 1/2
• Find the solutions in the given interval: x = π/6, 5π/6