Algebra Basics
| Name | Formula |
|---|---|
| Quadratic formula | x = (-b ± √(b²-4ac)) / 2a |
| Vieta's formulas | x₁ + x₂ = -b/a, x₁x₂ = c/a |
| Discriminant | Δ = b² - 4ac |
| Factoring | a² - b² = (a+b)(a-b) |
| Perfect square | (a±b)² = a² ± 2ab + b² |
| Sum/diff of cubes | a³ ± b³ = (a±b)(a² ∓ ab + b²) |
| Binomial theorem | (a+b)ⁿ = Σ C(n,k)aⁿ⁻ᵏbᵏ |
Sequences
| Name | Formula |
|---|---|
| Arithmetic sequence | aₙ = a₁ + (n-1)d |
| Arithmetic sum | Sₙ = n(a₁+aₙ)/2 = na₁ + n(n-1)d/2 |
| Geometric sequence | aₙ = a₁ · rⁿ⁻¹ |
| Geometric sum | Sₙ = a₁(1-rⁿ)/(1-r), r≠1 |
| Infinite geometric | S = a₁/(1-r), |r|<1 |
Functions
| Name | Formula |
|---|---|
| Exponent rules | aᵐ · aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ |
| Logarithm rules | log(xy) = logx + logy |
| Change of base | logₐb = logₓb / logₓa |
| Exp-log relation | y = aˣ ⟺ x = logₐy |
| Composite function | (f∘g)(x) = f(g(x)) |
| Inverse function | f(f⁻¹(x)) = x |
Trigonometry
| Name | Formula |
|---|---|
| Basic identity | sin²θ + cos²θ = 1 |
| Tangent | tanθ = sinθ/cosθ |
| Sum formula (sin) | sin(α±β) = sinαcosβ ± cosαsinβ |
| Sum formula (cos) | cos(α±β) = cosαcosβ ∓ sinαsinβ |
| Double angle (sin) | sin2θ = 2sinθcosθ |
| Double angle (cos) | cos2θ = cos²θ - sin²θ |
| Law of sines | a/sinA = b/sinB = c/sinC = 2R |
| Law of cosines | c² = a² + b² - 2ab·cosC |
Calculus
| Name | Formula |
|---|---|
| Derivative definition | f'(x) = lim[h→0] (f(x+h)-f(x))/h |
| Power rule | (xⁿ)' = nxⁿ⁻¹ |
| Exponential | (eˣ)' = eˣ, (aˣ)' = aˣlna |
| Logarithm | (lnx)' = 1/x |
| Trig derivatives | (sinx)' = cosx, (cosx)' = -sinx |
| Product rule | (fg)' = f'g + fg' |
| Quotient rule | (f/g)' = (f'g - fg')/g² |
| Chain rule | (f(g(x)))' = f'(g(x))·g'(x) |
Analytic Geometry
| Name | Formula |
|---|---|
| Distance formula | d = √((x₂-x₁)² + (y₂-y₁)²) |
| Midpoint formula | M = ((x₁+x₂)/2, (y₁+y₂)/2) |
| Slope | k = (y₂-y₁)/(x₂-x₁) |
| Point-slope form | y - y₁ = k(x - x₁) |
| General form | Ax + By + C = 0 |
| Point-line distance | d = |Ax₀+By₀+C| / √(A²+B²) |
| Circle equation | (x-a)² + (y-b)² = r² |
| Ellipse equation | x²/a² + y²/b² = 1 |
| Hyperbola equation | x²/a² - y²/b² = 1 |
| Parabola equation | y² = 4px 或 x² = 4py |
Vectors & Complex Numbers
| Name | Formula |
|---|---|
| Vector addition | a⃗ + b⃗ = (a₁+b₁, a₂+b₂) |
| Dot product | a⃗·b⃗ = |a⃗||b⃗|cosθ = a₁b₁+a₂b₂ |
| Vector magnitude | |a⃗| = √(a₁² + a₂²) |
| Angle between vectors | cosθ = a⃗·b⃗ / (|a⃗||b⃗|) |
| Complex addition | (a+bi) + (c+di) = (a+c) + (b+d)i |
| Complex multiplication | (a+bi)(c+di) = (ac-bd) + (ad+bc)i |
| Complex modulus | |a+bi| = √(a² + b²) |
Probability & Statistics
| Name | Formula |
|---|---|
| Classical probability | P(A) = n(A) / n(S) |
| Addition rule | P(A∪B) = P(A) + P(B) - P(A∩B) |
| Multiplication rule | P(A∩B) = P(A)·P(B|A) |
| Conditional prob. | P(B|A) = P(A∩B) / P(A) |
| Permutation | P(n,r) = n! / (n-r)! |
| Combination | C(n,r) = n! / (r!(n-r)!) |
| Mean | x̄ = Σxᵢ / n |
| Variance | s² = Σ(xᵢ-x̄)² / n |
| Standard deviation | s = √(s²) |