Essential Math Formulas and Definitions
Lines
Slope-Intercept Form
y = mx + b
Point-Slope Form
y – y₁ = m(x – x₁)
Standard Form
Ax + By = C
Rate of Change (Slope)
Δy/Δx = (y₁ – y₂)/(x₁ – x₂)
Algebra Formulas
Quadratic Equations
Standard Form
ax² + bx + c = 0
Quadratic Formula
x = [-b ± √(b² – 4ac)] / (2a)
Binomial Expansions
(a + b)³ Expansion
a³ + 3a²b + 3ab² + b³
(a – b)³ Expansion
a³ – 3a²b + 3ab² – b³
(a + b)² Expansion
a² + 2ab + b²
(a – b)² Expansion
a² – 2ab + b²
Exponent Rules
Product Rule
xᵃ · xᵇ = xᵃ⁺ᵇ
Quotient Rule
xᵃ / xᵇ = xᵃ⁻ᵇ
Power Rule
(xᵃ)ᵇ = xᵃᵇ
Negative Exponent
x⁻ᵃ = 1 / xᵃ
Logarithm Rules
Product Rule
logₐ(xy) = logₐ x + logₐ y
Quotient Rule
logₐ(x/y) = logₐ x – logₐ y
Power Rule
logₐ(xᵏ) = k logₐ x
Change of Base
logₐ b = ln b / ln a
Geometry Formulas
Sphere Formulas
Surface Area
SA = 4πr²
Volume
V = (4/3)πr³
Cone Formulas
Surface Area
SA = πr(r + h)
Volume
V = (1/3)πr²h
Trapezoid Area
A = ½(b₁ + b₂)h
Trigonometry Formulas
Right Triangle Ratios
Sine (sin θ)
opposite / hypotenuse
Cosine (cos θ)
adjacent / hypotenuse
Tangent (tan θ)
opposite / adjacent
Unit Circle Ratios
Sine (sin θ)
y/r
Cosine (cos θ)
x/r
Tangent (tan θ)
y/x
Reciprocal Identities
Cosecant (csc θ)
1 / sin θ
Secant (sec θ)
1 / cos θ
Cotangent (cot θ)
1 / tan θ
General Identity
sin θ · csc θ = 1 (and similarly for sec · cos, cot · tan)
Pythagorean Identities
Identity 1
sin² x + cos² x = 1
Identity 2
1 + tan² x = sec² x
Identity 3
1 + cot² x = csc² x
Double-Angle Identities
Sine (sin 2θ)
sin(2θ) = 2 sin θ cos θ
Cosine (cos 2θ)
cos(2θ) = cos² θ – sin² θ = 1 – 2 sin² θ = 2 cos² θ – 1
Sum-Difference Identities
Sine (sin(u±v))
sin(u±v) = sin u cos v ± cos u sin v
Cosine (cos(u±v))
cos(u±v) = cos u cos v ∓ sin u sin v
General Sinusoid Form
Sine Form
y = a sin[b(θ – c)] + d
Cosine Form
y = a cos[b(θ – c)] + d
Tangent Function Form
y = a tan[b(θ – c)] + d
Functions & Domains
Function Types
Even Function
f(–x) = f(x)
Odd Function
f(–x) = –f(x)
Domain Restrictions
Square Root
For √[s(x)], require s(x) ≥ 0.
Reciprocal
For 1/s(x), require s(x) ≠ 0.
Natural Logarithm
For ln[s(x)], require s(x) > 0.
Calculus Formulas
Limits
Limit sin(cx)/x
limₓ→0 [sin(cx)/x] = c
Limit (cos(cx)-1)/x
limₓ→0 [(cos(cx) – 1)/x] = 0
Continuity
Definition at x = c
limₓ→c f(x) = f(c)
Derivatives
Definition
f′(x) = limₕ→0 [f(x+h) – f(x)] / h
Product Rule
(uv)′ = u′v + uv′
Quotient Rule
(u/v)′ = (u′v – uv′) / v²
Chain Rule
d/dx [f(g(x))] = f′(g(x)) · g′(x)
Derivative ln u
(ln u)′ = u′ / u
Derivative eᵘ
(eᵘ)′ = u′ eᵘ
Derivative sin u
(sin u)′ = u′ cos u
Derivative cos u
(cos u)′ = –u′ sin u
Derivative tan u
(tan u)′ = u′ sec² u
Inverse Trig Derivatives
- arcsin u:
u′ / √(1 – u²) - arccos u:
–u′ / √(1 – u²) - arctan u:
u′ / (1 + u²)
Other Trig Derivatives
- d/dx [cot(u)] =
–csc²(u) · u′ - d/dx [sec(u)] =
sec(u) · tan(u) · u′ - d/dx [csc(u)] =
–csc(u) · cot(u) · u′
Second Derivative Notation
f″(x) or d²y/dx²
Inverse Function Derivative
(f⁻¹)′(x) = 1 / f′(f⁻¹(x))
Theorems
Rolle’s Theorem & Mean Value Theorem
Key calculus theorems.
L’Hôpital’s Rule
lim f/g = f′/g′ under 0/0 or ∞/∞ indeterminate forms.
Integrals
Definite Integral (Riemann Sum)
∫ₐᵇ f(x) dx = limn→∞ Σk f(xk) · Δx
Analytic Geometry
Asymptotes
Oblique Asymptotes
Occur when degree of numerator = degree of denominator + 1. Find using polynomial division.
Conic Sections
- Circle:
(x–h)² + (y–k)² = r² - Ellipse:
(x–h)²/a² + (y–k)²/b² = 1 - Parabola (Vertical):
(x–h)² = 4p(y–k) - Parabola (Horizontal):
(y–k)² = 4p(x–h) - Hyperbola:
(x–h)²/a² – (y–k)²/b² = 1
Coordinate Conversions
Polar to Rectangular
x = r cos θ, y = r sin θ
Rectangular to Polar
r = √(x² + y²), θ = arctan(y/x)
Linear Algebra (2×2)
Matrix Formulas
Determinant
For matrix [[a, b], [c, d]], det(A) = ad – bc
Inverse
For matrix [[a, b], [c, d]], A⁻¹ = (1/det(A)) · [[d, –b], [–c, a]]
Complex Numbers
Trigonometric Form
z = r(cos θ + i sin θ)