{
 "source": "https://peabody.kids/flashcards/math-trig-identities",
 "id": "math-trig-identities",
 "title": "Trig Definitions and Identities",
 "category": "math",
 "subject": "Math",
 "levels": [
  "9-12",
  "college"
 ],
 "description": "Right-triangle ratios, reciprocal, Pythagorean, angle-sum, double-angle and half-angle identities.",
 "tags": [
  "trigonometry",
  "identities"
 ],
 "frontLabel": "Name",
 "backLabel": "Identity",
 "cards": [
  [
   "SOH-CAH-TOA: sine",
   "sin = opposite / hypotenuse"
  ],
  [
   "SOH-CAH-TOA: cosine",
   "cos = adjacent / hypotenuse"
  ],
  [
   "SOH-CAH-TOA: tangent",
   "tan = opposite / adjacent"
  ],
  [
   "Cosecant",
   "csc θ = 1 / sin θ"
  ],
  [
   "Secant",
   "sec θ = 1 / cos θ"
  ],
  [
   "Cotangent",
   "cot θ = 1 / tan θ = cos θ / sin θ"
  ],
  [
   "Tangent in terms of sine and cosine",
   "tan θ = sin θ / cos θ"
  ],
  [
   "Pythagorean identity",
   "sin² θ + cos² θ = 1"
  ],
  [
   "Pythagorean identity with tangent",
   "1 + tan² θ = sec² θ"
  ],
  [
   "Pythagorean identity with cotangent",
   "1 + cot² θ = csc² θ"
  ],
  [
   "Even-odd: sin(−θ)",
   "−sin θ (sine is odd)"
  ],
  [
   "Even-odd: cos(−θ)",
   "cos θ (cosine is even)"
  ],
  [
   "Even-odd: tan(−θ)",
   "−tan θ (tangent is odd)"
  ],
  [
   "Cofunction: sin(90° − θ)",
   "cos θ"
  ],
  [
   "Cofunction: cos(90° − θ)",
   "sin θ"
  ],
  [
   "Cofunction: tan(90° − θ)",
   "cot θ"
  ],
  [
   "Supplementary: sin(180° − θ)",
   "sin θ"
  ],
  [
   "Supplementary: cos(180° − θ)",
   "−cos θ"
  ],
  [
   "Shift: sin(θ + 180°)",
   "−sin θ"
  ],
  [
   "Shift: cos(θ + 180°)",
   "−cos θ"
  ],
  [
   "Shift: tan(θ + 180°)",
   "tan θ"
  ],
  [
   "Sum formula for sine",
   "sin(A + B) = sin A cos B + cos A sin B"
  ],
  [
   "Difference formula for sine",
   "sin(A − B) = sin A cos B − cos A sin B"
  ],
  [
   "Sum formula for cosine",
   "cos(A + B) = cos A cos B − sin A sin B"
  ],
  [
   "Difference formula for cosine",
   "cos(A − B) = cos A cos B + sin A sin B"
  ],
  [
   "Sum formula for tangent",
   "tan(A + B) = (tan A + tan B) / (1 − tan A tan B)"
  ],
  [
   "Difference formula for tangent",
   "tan(A − B) = (tan A − tan B) / (1 + tan A tan B)"
  ],
  [
   "Double-angle: sine",
   "sin 2θ = 2 sin θ cos θ"
  ],
  [
   "Double-angle: cosine",
   "cos 2θ = cos² θ − sin² θ"
  ],
  [
   "Double-angle: cosine (other form 1)",
   "cos 2θ = 1 − 2 sin² θ"
  ],
  [
   "Double-angle: cosine (other form 2)",
   "cos 2θ = 2 cos² θ − 1"
  ],
  [
   "Double-angle: tangent",
   "tan 2θ = 2 tan θ / (1 − tan² θ)"
  ],
  [
   "Power reducing: sin² θ",
   "(1 − cos 2θ) / 2"
  ],
  [
   "Power reducing: cos² θ",
   "(1 + cos 2θ) / 2"
  ],
  [
   "Half-angle: sin(θ/2)",
   "±√((1 − cos θ) / 2)"
  ],
  [
   "Half-angle: cos(θ/2)",
   "±√((1 + cos θ) / 2)"
  ],
  [
   "Product to sum: sin A cos B",
   "½[sin(A + B) + sin(A − B)]"
  ],
  [
   "Product to sum: cos A cos B",
   "½[cos(A − B) + cos(A + B)]"
  ],
  [
   "Law of sines",
   "a / sin A = b / sin B = c / sin C"
  ],
  [
   "Law of cosines",
   "c² = a² + b² − 2ab cos C"
  ],
  [
   "Area of a triangle with trig",
   "½ ab sin C"
  ],
  [
   "Arc length formula (radians)",
   "s = rθ"
  ],
  [
   "Sector area formula (radians)",
   "A = ½ r² θ"
  ],
  [
   "Degrees to radians",
   "Multiply by π/180"
  ],
  [
   "Radians to degrees",
   "Multiply by 180/π"
  ],
  [
   "Amplitude of y = A sin(Bx)",
   "|A|"
  ],
  [
   "Period of y = A sin(Bx)",
   "2π / |B|"
  ],
  [
   "Phase shift of y = sin(x − C)",
   "C units to the right"
  ],
  [
   "Inverse sine range",
   "−π/2 to π/2"
  ],
  [
   "Inverse cosine range",
   "0 to π"
  ],
  [
   "Inverse tangent range",
   "−π/2 to π/2 (not including the ends)"
  ],
  [
   "Euler's formula",
   "e^(iθ) = cos θ + i sin θ"
  ],
  [
   "Euler's identity",
   "e^(iπ) + 1 = 0"
  ],
  [
   "De Moivre's theorem",
   "(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ"
  ]
 ]
}