Class 11–12 — Maths Formula Sheet

sin(90°−θ) = cosθ
cos(90°−θ) = sinθ
tan(90°−θ) = cotθ

Trig ratios for complementary angles.

sin(A±B) = sinA cosB ± cosA sinB
cos(A±B) = cosA cosB ∓ sinA sinB
tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB)

Expand trig functions of sum or difference of angles.

sin 2A = 2 sinA cosA
cos 2A = cos²A − sin²A = 2cos²A−1 = 1−2sin²A
tan 2A = 2tanA/(1−tan²A)

Express trig of double angle in terms of single angle.

sin²(A/2) = (1−cosA)/2
cos²(A/2) = (1+cosA)/2
tan²(A/2) = (1−cosA)/(1+cosA)

Express half-angle in terms of full angle.

2sinA cosB = sin(A+B)+sin(A−B)
2cosA cosB = cos(A+B)+cos(A−B)
2sinA sinB = cos(A−B)−cos(A+B)

Convert product of trig functions to sum.

aₙ = arⁿ⁻¹

General term of geometric progression.

📌 a = first term, r = common ratio

Sₙ = a(rⁿ−1)/(r−1)  if r ≠ 1
Sₙ = na  if r = 1

Sum of first n terms of a GP.

S∞ = a/(1−r),  |r| < 1

Sum of all terms of convergent GP.

(a+b)/2 ≥ √(ab)  for a,b > 0

Arithmetic mean is always ≥ geometric mean.

Σn = n(n+1)/2

Sum 1+2+3+…+n.

Σn² = n(n+1)(2n+1)/6

Sum 1²+2²+3²+…+n².

Σn³ = [n(n+1)/2]²

Sum 1³+2³+3³+…+n³.

n! = n × (n−1) × … × 2 × 1
0! = 1

Product of all positive integers up to n.

ⁿPᵣ = n! / (n−r)!

Ordered arrangements of r items from n.

💡 ⁵P₃ = 5!/2! = 60

ⁿCᵣ = n! / [r!(n−r)!]

Unordered selections of r items from n.

💡 ⁵C₃ = 5!/(3!2!) = 10

(a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ  (r=0 to n)

Expansion of binomial raised to power n.

Tᵣ₊₁ = ⁿCᵣ × aⁿ⁻ʳ × bʳ

(r+1)th term of the binomial expansion.

lim(x→0) sinx/x = 1

Important limit used in differentiation from first principles.

lim(x→0) (1+x)^(1/x) = e

Definition of Euler's number e ≈ 2.718.

lim(x→a) (xⁿ−aⁿ)/(x−a) = naⁿ⁻¹

Algebraic limit formula.

d/dx(xⁿ) = nxⁿ⁻¹

Differentiate x raised to any power.

💡 d/dx(x³) = 3x²

d/dx(sinx) = cosx
d/dx(cosx) = −sinx

Basic derivatives of trigonometric functions.

d/dx[f(g(x))] = f'(g(x)) × g'(x)

Differentiate composite functions.

d/dx(uv) = u'v + uv'

Differentiate a product of two functions.

d/dx(u/v) = (u'v − uv') / v²

Differentiate a quotient of two functions.

∫xⁿ dx = xⁿ⁺¹/(n+1) + C  (n ≠ −1)

Integrate x to any power (except −1).

∫sinx dx = −cosx + C
∫cosx dx = sinx + C

Basic integrals of sin and cos.

a⃗ · b⃗ = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃

Scalar product of two vectors.

💡 Angle between vectors: θ = cos⁻¹(a⃗·b⃗/|a||b|)

|a⃗ × b⃗| = |a||b|sinθ

Magnitude of vector product; useful for area of parallelogram.

d = √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]

Distance between two points in 3D space.

l²+m²+n² = 1
l = cosα, m = cosβ, n = cosγ

Cosines of angles a line makes with x, y, z axes.

σ² = Σ(xᵢ−x̄)²/n

Population variance — average of squared deviations.

σ = √[Σ(xᵢ−x̄)²/n]

Square root of variance; same unit as data.

CV = (σ/x̄) × 100

Relative measure of dispersion; useful for comparing datasets.

P(A|B) = P(B|A)×P(A) / P(B)

Conditional probability — updating probability given new evidence.