Trigonometric identities

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Fundamental Identities

Pythagorean Identities

  • sin²θ + cos²θ = 1
  • tan²θ + 1 = sec²θ
  • 1 + cot²θ = csc²θ

Reciprocal Identities

  • sin θ = 1/csc θ
  • cos θ = 1/sec θ
  • tan θ = 1/cot θ
  • csc θ = 1/sin θ
  • sec θ = 1/cos θ
  • cot θ = 1/tan θ

Quotient Identities

  • tan θ = sin θ/cos θ
  • cot θ = cos θ/sin θ

Compound Angle Formulas

Sum and Difference Formulas

  • sin(α ± β) = sin α cos β ± cos α sin β
  • cos(α ± β) = cos α cos β ∓ sin α sin β
  • tan(α ± β) = (tan α ± tan β)/(1 ∓ tan α tan β)

Double Angle Formulas

  • sin(2θ) = 2sin θ cos θ
  • cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
  • tan(2θ) = 2tan θ/(1 - tan²θ)

Triple Angle Formulas

  • sin(3θ) = 3sin θ - 4sin³θ
  • cos(3θ) = 4cos³θ - 3cos θ

Half Angle Formulas

  • sin(θ/2) = ±√[(1 - cos θ)/2] (sign depends on quadrant)
  • cos(θ/2) = ±√[(1 + cos θ)/2] (sign depends on quadrant)
  • tan(θ/2) = (1 - cos θ)/sin θ = sin θ/(1 + cos θ)

Product to Sum Formulas

  • sin α sin β = (1/2)[cos(α - β) - cos(α + β)]
  • cos α cos β = (1/2)[cos(α - β) + cos(α + β)]
  • sin α cos β = (1/2)[sin(α + β) + sin(α - β)]

Sum to Product Formulas

  • sin α + sin β = 2sin((α + β)/2)cos((α - β)/2)
  • sin α - sin β = 2cos((α + β)/2)sin((α - β)/2)
  • cos α + cos β = 2cos((α + β)/2)cos((α - β)/2)
  • cos α - cos β = -2sin((α + β)/2)sin((α - β)/2)

Power Reduction Formulas

  • sin²θ = (1 - cos(2θ))/2
  • cos²θ = (1 + cos(2θ))/2
  • sin³θ = (3sin θ - sin(3θ))/4
  • cos³θ = (3cos θ + cos(3θ))/4

Application Strategies

For Proving Identities

  1. Start with the more complex side
  2. Use fundamental identities to simplify
  3. Transform to look like the other side
  4. Avoid working with both sides simultaneously

For Solving Trigonometric Equations

  1. Express everything in terms of a single function (e.g., all in sin θ)
  2. Apply algebraic methods (factoring, completing square, etc.)
  3. Find all solutions within the principal interval
  4. Extend to full domain if needed

Example: Showing 4sin²x cos²x = (1-cos(4x))/2

  1. Use sin(2x) = 2sin x cos x
  2. 4sin²x cos²x = (2sin x cos x)²
  3. = sin²(2x)
  4. Apply sin²θ = (1 - cos(2θ))/2
  5. = (1 - cos(4x))/2

Example: Solving sin x cos x + 1 = 0

  1. Use sin(2x) = 2sin x cos x
  2. sin(2x)/2 + 1 = 0
  3. sin(2x) = -2
  4. Since sin θ ≤ 1, this has no solution

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