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

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