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Trigonometric Identities & Equations

60 cards · 🇺🇸 English

Core trig identities and solving trigonometric equations.

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Trigonometric identity

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  1. 1Trigonometric identity
    An equality involving trigonometric functions that is true for every value of the variables
  2. 2Fundamental Pythagorean identity
    sin²θ + cos²θ = 1
  3. 3Derive the Pythagorean identity from the unit circle
    On the unit circle, x² + y² = 1, where x = cos θ and y = sin θ, hence cos²θ + sin²θ = 1
  4. 4Second Pythagorean identity
    1 + tan²θ = sec²θ
  5. 5Third Pythagorean identity
    1 + cot²θ = csc²θ
  6. 6Derive 1 + tan²θ = sec²θ
    Divide sin²θ + cos²θ = 1 by cos²θ: (sin²θ/cos²θ) + 1 = 1/cos²θ, thus tan²θ + 1 = sec²θ
  7. 7Sin²θ in terms of cos θ
    sin²θ = 1 − cos²θ
  8. 8Cos²θ in terms of sin θ
    cos²θ = 1 − sin²θ
  9. 9Reciprocal identity for sine
    csc θ = 1/sin θ
  10. 10Reciprocal identity for cosine
    sec θ = 1/cos θ
  11. 11Reciprocal identity for tangent
    cot θ = 1/tan θ
  12. 12Quotient identity for tangent
    tan θ = sin θ / cos θ
  13. 13Quotient identity for cotangent
    cot θ = cos θ / sin θ
  14. 14Even/odd property of sine
    sin(−θ) = −sin θ (sine is an odd function)
  15. 15Even/odd property of cosine
    cos(−θ) = cos θ (cosine is an even function)
  16. 16Even/odd property of tangent
    tan(−θ) = −tan θ (tangent is an odd function)
  17. 17Cofunction identity for sine
    sin(π/2 − θ) = cos θ
  18. 18Cofunction identity for cosine
    cos(π/2 − θ) = sin θ
  19. 19Cofunction identity for tangent
    tan(π/2 − θ) = cot θ
  20. 20Period of sine and cosine
    2π (360°) — sin(θ + 2π) = sin θ and cos(θ + 2π) = cos θ
  21. 21Period of tangent and cotangent
    π (180°) — tan(θ + π) = tan θ and cot(θ + π) = cot θ
  22. 22Angle sum formula for sine
    sin(α + β) = sin α cos β + cos α sin β
  23. 23Angle sum formula for cosine
    cos(α + β) = cos α cos β − sin α sin β
  24. 24Angle sum formula for tangent
    tan(α + β) = (tan α + tan β) / (1 − tan α tan β)
  25. 25Angle difference formula for sine
    sin(α − β) = sin α cos β − cos α sin β
  26. 26Angle difference formula for cosine
    cos(α − β) = cos α cos β + sin α sin β
  27. 27Angle difference formula for tangent
    tan(α − β) = (tan α − tan β) / (1 + tan α tan β)
  28. 28Double-angle formula for sine
    sin(2θ) = 2 sin θ cos θ
  29. 29Double-angle formula for cosine (first form)
    cos(2θ) = cos²θ − sin²θ
  30. 30Double-angle formula for cosine (second form)
    cos(2θ) = 2cos²θ − 1

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