Mathematical Formula Cheat Sheet & Identities

A comprehensive reference library of standard algebraic, trigonometric, logarithmic, and combinatoric identities for students and professionals.

Pythagorean Identities

sin²(θ) + cos²(θ) = 1
1 + tan²(θ) = sec²(θ)
1 + cot²(θ) = csc²(θ)

Angle Addition & Subtraction

sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)
tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β)

Double Angle Formulas

sin(2θ) = 2 sin(θ) cos(θ)
cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1
tan(2θ) = 2 tan(θ) / (1 - tan²(θ))

Product & Quotient Rules

log_b(xy) = log_b(x) + log_b(y)
log_b(x / y) = log_b(x) - log_b(y)

Power & Change of Base

log_b(x^k) = k × log_b(x)
log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)

Exponential Properties

xᵃ × xᵇ = x^(a + b)
xᵃ / xᵇ = x^(a - b)
(xᵃ)ᵇ = x^(a × b)

Factorials (n!)

n! = n × (n - 1) × (n - 2) × ... × 1
0! = 1 | 1! = 1

Permutations P(n, r)

P(n, r) = n! / (n - r)!
Ordered arrangements of r items chosen from n distinct items.

Combinations C(n, r)

C(n, r) = n! / (r! × (n - r)!)
Unordered selections of r items chosen from n distinct items.