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Trig Definitions and Identities

Right-triangle ratios, reciprocal, Pythagorean, angle-sum, double-angle and half-angle identities.

54 cardsGrades 9–12, College#trigonometry#identities

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54 cards
  • SOH-CAH-TOA: sineName
    sin = opposite / hypotenuse
  • SOH-CAH-TOA: cosineName
    cos = adjacent / hypotenuse
  • SOH-CAH-TOA: tangentName
    tan = opposite / adjacent
  • CosecantName
    csc θ = 1 / sin θ
  • SecantName
    sec θ = 1 / cos θ
  • CotangentName
    cot θ = 1 / tan θ = cos θ / sin θ
  • Tangent in terms of sine and cosineName
    tan θ = sin θ / cos θ
  • Pythagorean identityName
    sin² θ + cos² θ = 1
  • Pythagorean identity with tangentName
    1 + tan² θ = sec² θ
  • Pythagorean identity with cotangentName
    1 + cot² θ = csc² θ
  • Even-odd: sin(−θ)Name
    −sin θ (sine is odd)
  • Even-odd: cos(−θ)Name
    cos θ (cosine is even)
  • Even-odd: tan(−θ)Name
    −tan θ (tangent is odd)
  • Cofunction: sin(90° − θ)Name
    cos θ
  • Cofunction: cos(90° − θ)Name
    sin θ
  • Cofunction: tan(90° − θ)Name
    cot θ
  • Supplementary: sin(180° − θ)Name
    sin θ
  • Supplementary: cos(180° − θ)Name
    −cos θ
  • Shift: sin(θ + 180°)Name
    −sin θ
  • Shift: cos(θ + 180°)Name
    −cos θ
  • Shift: tan(θ + 180°)Name
    tan θ
  • Sum formula for sineName
    sin(A + B) = sin A cos B + cos A sin B
  • Difference formula for sineName
    sin(A − B) = sin A cos B − cos A sin B
  • Sum formula for cosineName
    cos(A + B) = cos A cos B − sin A sin B
  • Difference formula for cosineName
    cos(A − B) = cos A cos B + sin A sin B
  • Sum formula for tangentName
    tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
  • Difference formula for tangentName
    tan(A − B) = (tan A − tan B) / (1 + tan A tan B)
  • Double-angle: sineName
    sin 2θ = 2 sin θ cos θ
  • Double-angle: cosineName
    cos 2θ = cos² θ − sin² θ
  • Double-angle: cosine (other form 1)Name
    cos 2θ = 1 − 2 sin² θ
  • Double-angle: cosine (other form 2)Name
    cos 2θ = 2 cos² θ − 1
  • Double-angle: tangentName
    tan 2θ = 2 tan θ / (1 − tan² θ)
  • Power reducing: sin² θName
    (1 − cos 2θ) / 2
  • Power reducing: cos² θName
    (1 + cos 2θ) / 2
  • Half-angle: sin(θ/2)Name
    ±√((1 − cos θ) / 2)
  • Half-angle: cos(θ/2)Name
    ±√((1 + cos θ) / 2)
  • Product to sum: sin A cos BName
    ½[sin(A + B) + sin(A − B)]
  • Product to sum: cos A cos BName
    ½[cos(A − B) + cos(A + B)]
  • Law of sinesName
    a / sin A = b / sin B = c / sin C
  • Law of cosinesName
    c² = a² + b² − 2ab cos C
  • Area of a triangle with trigName
    ½ ab sin C
  • Arc length formula (radians)Name
    s = rθ
  • Sector area formula (radians)Name
    A = ½ r² θ
  • Degrees to radiansName
    Multiply by π/180
  • Radians to degreesName
    Multiply by 180/π
  • Amplitude of y = A sin(Bx)Name
    |A|
  • Period of y = A sin(Bx)Name
    2π / |B|
  • Phase shift of y = sin(x − C)Name
    C units to the right
  • Inverse sine rangeName
    −π/2 to π/2
  • Inverse cosine rangeName
    0 to π
  • Inverse tangent rangeName
    −π/2 to π/2 (not including the ends)
  • Euler's formulaName
    e^(iθ) = cos θ + i sin θ
  • Euler's identityName
    e^(iπ) + 1 = 0
  • De Moivre's theoremName
    (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ

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