Mathematics Formula Sheet
90 formulas across 10 chapters — with variables explained and exam tips where needed.
Ch 3Trigonometric Functions(12 formulas)
Pythagorean identity 1
sin²θ + cos²θ = 1
Pythagorean identity 2
1 + tan²θ = sec²θ
Pythagorean identity 3
1 + cot²θ = cosec²θ
Sine of sum
sin(A+B) = sinA cosB + cosA sinB
Remember as: sin sum = sin·cos + cos·sin
Cosine of sum
cos(A+B) = cosA cosB − sinA sinB
Tangent of sum
tan(A+B) = (tanA + tanB) / (1 − tanA tanB)
Double angle: sine
sin2A = 2 sinA cosA
Double angle: cosine
cos2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1
Double angle: tangent
tan2A = 2tanA / (1 − tan²A)
Product to sum
2 sinA cosB = sin(A+B) + sin(A−B)
Sum to product
sinC + sinD = 2 sin[(C+D)/2] cos[(C+D)/2]
Useful for simplifying sums of trig functions
Principal values
sin⁻¹: [−π/2, π/2] | cos⁻¹: [0, π] | tan⁻¹: (−π/2, π/2)
Ch 4Complex Numbers and Quadratic Equations(8 formulas)
Complex number definition
z = a + ib
a = real part, b = imaginary part, i = √(−1)
Modulus of z
|z| = √(a² + b²)
Argument of z
arg(z) = θ = tan⁻¹(b/a) (adjusted for correct quadrant)
Polar form
z = r(cosθ + i sinθ) = re^(iθ)
r = |z|, θ = arg(z)
De Moivre's theorem
(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)
Conjugate
z̄ = a − ib | z × z̄ = |z|²
Multiply by conjugate to rationalise complex fractions
Quadratic discriminant
D = b² − 4ac
D > 0: two real roots; D = 0: equal real roots; D < 0: complex conjugate roots
Complex roots occur in pairs
If p + iq is a root of real-coefficient polynomial, then p − iq is also a root
Ch 7Permutations and Combinations(7 formulas)
Permutation (arrangement)
P(n,r) = nPr = n! / (n−r)!
n = total items, r = items selected
Order matters in permutation
Combination (selection)
C(n,r) = nCr = n! / [r!(n−r)!]
Order does NOT matter in combination
Circular permutations
(n−1)! (for n distinct items in a circle)
If clockwise and anticlockwise are same: (n−1)!/2
Permutations with repetition
n!/p!q!r!...
p, q, r = counts of repeated items
Symmetry of nCr
nCr = nC(n−r)
Pascal's identity
nCr + nC(r+1) = (n+1)C(r+1)
Sum of all nCr
nC0 + nC1 + ... + nCn = 2ⁿ
Ch 8Binomial Theorem(6 formulas)
Binomial expansion
(a + b)ⁿ = Σ(r=0 to n) nCr × a^(n−r) × bʳ
General term
T_(r+1) = nCr × a^(n−r) × bʳ
To find term containing bᵏ, set r = k
Middle term (n even)
T_(n/2 + 1) is the single middle term
Middle terms (n odd)
T_((n+1)/2) and T_((n+3)/2) are both middle terms
Sum of coefficients
Put a = b = 1: sum = 2ⁿ
Sum of odd-position coefficients
nC0 + nC2 + nC4 + ... = 2^(n−1)
Same for even-position coefficients
Ch 9Sequences and Series(11 formulas)
AP: nth term
aₙ = a + (n−1)d
a = first term, d = common difference
AP: sum of n terms
Sₙ = n/2 × [2a + (n−1)d] = n/2 × (a + l)
l = last term
Use second form when first and last terms are known
GP: nth term
aₙ = ar^(n−1)
r = common ratio
GP: sum of n terms (r ≠ 1)
Sₙ = a(rⁿ − 1) / (r − 1) for r > 1 | a(1 − rⁿ)/(1 − r) for r < 1
GP: sum of infinite series (|r| < 1)
S∞ = a / (1 − r)
Arithmetic mean
AM of a and b = (a + b) / 2
Geometric mean
GM of a and b = √(ab)
AM-GM inequality
AM ≥ GM (equality when all terms are equal)
Sum of first n natural numbers
Σn = n(n+1)/2
Sum of squares
Σn² = n(n+1)(2n+1)/6
Sum of cubes
Σn³ = [n(n+1)/2]²
Σn³ = (Σn)² — a beautiful result worth memorising
Ch 10Straight Lines(9 formulas)
Slope formula
m = (y₂ − y₁) / (x₂ − x₁) = tanθ
θ = inclination of line with positive x-axis
Slope-intercept form
y = mx + c
c = y-intercept
Point-slope form
y − y₁ = m(x − x₁)
Two-point form
(y − y₁) / (y₂ − y₁) = (x − x₁) / (x₂ − x₁)
Intercept form
x/a + y/b = 1
a = x-intercept, b = y-intercept
Normal form
x cosω + y sinω = p
p = perpendicular distance from origin, ω = angle normal makes with x-axis
General form
ax + by + c = 0
Distance from point (x₁,y₁) to line ax+by+c=0
d = |ax₁ + by₁ + c| / √(a² + b²)
Angle between two lines
tanθ = |(m₁ − m₂) / (1 + m₁m₂)|
Lines parallel: m₁ = m₂; perpendicular: m₁m₂ = −1
Ch 11Conic Sections(9 formulas)
Circle: standard form
(x − h)² + (y − k)² = r²
Centre (h,k), radius r
Circle: general form
x² + y² + 2gx + 2fy + c = 0 | Centre: (−g, −f), r = √(g²+f²−c)
Parabola: standard
y² = 4ax | Focus: (a,0), Directrix: x = −a, Axis: x-axis
a > 0 → opens right; for y² = −4ax → opens left
Ellipse: standard
x²/a² + y²/b² = 1 (a > b) | c² = a² − b², Eccentricity e = c/a < 1
Ellipse: foci
Foci at (±c, 0) | Length of major axis = 2a | Minor axis = 2b
Hyperbola: standard
x²/a² − y²/b² = 1 | c² = a² + b², e = c/a > 1
Hyperbola: asymptotes
y = ±(b/a)x
Asymptotes pass through centre and are NOT part of the curve
Parabola: parametric form
x = at², y = 2at for y² = 4ax
Ellipse: parametric form
x = a cosθ, y = b sinθ
Ch 12Limits and Derivatives(12 formulas)
Standard limit 1
lim(x→0) sinx/x = 1
x must be in radians
Standard limit 2
lim(x→0) (1 − cosx)/x = 0
Standard limit 3
lim(x→0) tanx/x = 1
Standard limit 4
lim(x→a) (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹
Derivative from first principles
f'(x) = lim(h→0) [f(x+h) − f(x)] / h
Power rule
d/dx(xⁿ) = nxⁿ⁻¹
Product rule
d/dx(uv) = u·v' + v·u'
Quotient rule
d/dx(u/v) = (v·u' − u·v') / v²
d/dx(sin x)
cos x
d/dx(cos x)
−sin x
d/dx(tan x)
sec²x
d/dx(constant)
0
A very common mistake is forgetting that the derivative of a constant is zero
Ch 13Statistics(8 formulas)
Mean (ungrouped)
x̄ = Σxᵢ / n
Mean (grouped)
x̄ = Σfᵢxᵢ / Σfᵢ
fᵢ = frequency, xᵢ = class midpoint
Variance (ungrouped)
σ² = Σ(xᵢ − x̄)² / n = Σxᵢ²/n − x̄²
Variance (grouped)
σ² = Σfᵢ(xᵢ − x̄)² / Σfᵢ
Standard deviation
same as data unitsσ = √(σ²)
Coefficient of variation
%CV = (σ / x̄) × 100
CV is dimensionless — used to compare variability of two datasets with different units or means
Range
Range = Maximum value − Minimum value
Mean deviation from mean
MD(x̄) = Σ|xᵢ − x̄| / n
Ch 14Probability(8 formulas)
Classical probability
P(A) = n(A) / n(S)
n(A) = favourable outcomes, n(S) = total outcomes
Valid only when all outcomes are equally likely
Addition theorem
P(A∪B) = P(A) + P(B) − P(A∩B)
Mutually exclusive events
P(A∪B) = P(A) + P(B) when A∩B = ∅
Complement rule
P(A') = 1 − P(A)
Conditional probability
P(A|B) = P(A∩B) / P(B)
Read as: probability of A given that B has already occurred
Multiplication rule
P(A∩B) = P(A) × P(B|A) = P(B) × P(A|B)
Independent events
P(A∩B) = P(A) × P(B)
If events are independent: P(A|B) = P(A)
Bayes' theorem
P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σⱼ P(Aⱼ)P(B|Aⱼ)
Used when multiple causes can lead to the same observed event