IB maths formula booklet: AA and AI, SL and HL

Q: Is the formula booklet given in the IB maths exam?

The booklet's cover says it is for use during the course and in the examinations. Your school should supply a copy for the exam, and you should practise with it all year.

Q: Is the AA formula booklet the same for SL and HL?

Yes, AA SL and AA HL share one booklet, and the contents page splits each topic into SL and HL sections. AI SL and AI HL likewise share one booklet, separate from AA.

Q: Does the AI booklet print the quadratic formula?

Only in its HL part of the prior learning section. The AA booklet prints it for both SL and HL. AI SL students usually solve quadratics with the calculator, but learning the formula is still sensible.

Q: Why is my row marked AA only or AI only?

The two courses teach different content: AA prints more trigonometric identities, vector geometry and calculus, while AI includes matrices, Poisson, transition matrices, differential systems and the trapezoidal rule. Each row names the booklets that print it.

Q: What do I need to memorise that is not in the booklet?

Use the recall list on this page. It covers results such as laws of exponents, exact trigonometric values, outlier rules and tangent equations. Confirm the exact syllabus wording in your course guide with your teacher.

This page covers IB Diploma Mathematics: analysis and approaches (AA) and applications and interpretation (AI), at SL and HL, based on version 1.0 of the formula booklets (first examinations 2021). It is CuratedPrep's own organised list of the formulas, not a copy of the booklet, so read each row's note to see which course and level prints it, then work through the list of results you must recall.

Board	Official document
IB Mathematics: analysis and approaches (SL and HL)	Mathematics: analysis and approaches formula booklet, version 1.0
IB Mathematics: applications and interpretation (SL and HL)	Mathematics: applications and interpretation formula booklet, version 1.0

In the booklet means the formula is printed in the official booklet or sheet in the exam; Learn it means students must recall it.

Number and algebra

Sequences, finance, logarithms, binomial, complex numbers and matrices. AI SL prints only sequences, compound interest, the exponent and logarithm link, and percentage error.

Formula	Equation	In the exam
nth term of an arithmetic sequenceAA and AI, SL and HL. u₁ is the first term, d the common difference.In the booklet	$u_{n} = u_{1} + (n - 1) d$	In the booklet
Sum of n terms of an arithmetic sequenceAA and AI, SL and HL. Use the second form when you know the last term.In the booklet	$S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d) = \frac{n}{2} (u_{1} + u_{n})$	In the booklet
nth term of a geometric sequenceAA and AI, SL and HL. r is the common ratio.In the booklet	$u_{n} = u_{1} r^{n - 1}$	In the booklet
Sum of n terms of a finite geometric sequenceAA and AI, SL and HL. Pick the form that keeps the denominator positive.In the booklet	$S_{n} = \frac{u _{1} ( r ^{n} - 1 )}{r - 1} = \frac{u _{1} ( 1 - r ^{n} )}{1 - r}, r \neq = 1$	In the booklet
Sum to infinity of a geometric sequenceAA SL and HL; AI HL only (not printed for AI SL). Version 1.0 of the AA booklet shows the denominator as r minus 1, which is a misprint: use 1 minus r.In the booklet	$S_{\infty} = \frac{u _{1}}{1 - r}, ∣ r ∣ < 1$	In the booklet
Compound interestAA and AI, SL and HL. FV future value, PV present value, n years, k compounding periods per year, r% nominal annual rate.In the booklet	$F V = P V \times (1 + \frac{r}{100 k})^{kn}$	In the booklet
Link between exponents and logarithmsAA and AI, SL and HL.In the booklet	$a^{x} = b ⟺ x = lo g_{a} b, a > 0, b > 0, a \neq = 1$	In the booklet
Laws of logarithmsAA SL and HL; AI HL only (not printed for AI SL). Needs a, x, y > 0.In the booklet	$lo g_{a} x y = lo g_{a} x + lo g_{a} y; lo g_{a} \frac{x}{y} = lo g_{a} x - lo g_{a} y; lo g_{a} x^{m} = m lo g_{a} x$	In the booklet
Change of baseAA SL and HL only; not printed for AI.In the booklet	$lo g_{a} x = \frac{l o g _{b} x}{l o g _{b} a}$	In the booklet
Percentage errorAI SL and HL only; not printed for AA. v_E exact value, v_A approximate value.In the booklet	$ε = \frac{v _{A} - v _{E}}{v _{E}} \times 100%$	In the booklet
Binomial theoremAA SL and HL only; not printed for AI.In the booklet	$(a + b)^{n} = a^{n} + (1 n) a^{n - 1} b + \dots + (r n) a^{n - r} b^{r} + \dots + b^{n}$	In the booklet
Binomial coefficient (combinations)AA SL and HL (printed with the binomial theorem, and again in the HL combinations entry); not printed for AI.In the booklet	$(r n) = \frac{n !}{r ! ( n - r )!}$	In the booklet
PermutationsAA HL only.In the booklet	$^{n} P_{r} = \frac{n !}{( n - r )!}$	In the booklet
Complex number in Cartesian formAA HL and AI HL only.In the booklet	$z = a + bi$	In the booklet
Polar and Euler form of a complex numberAA HL and AI HL only. r is the modulus, θ the argument.In the booklet	$z = r (cos θ + i sin θ) = r e^{i θ} = r cis θ$	In the booklet
De Moivre's theoremAA HL only.In the booklet	$[r (cos θ + i sin θ)]^{n} = r^{n} (cos n θ + i sin n θ) = r^{n} e^{in θ}$	In the booklet
Determinant of a 2 by 2 matrixAI HL only.In the booklet	$A = (a c b d) \Rightarrow det A = a d - b c$	In the booklet
Inverse of a 2 by 2 matrixAI HL only.In the booklet	$A^{- 1} = \frac{1}{d e t A} (d - c - b a), a d \neq = b c$	In the booklet
Power of a matrix by diagonalisationAI HL only. P holds the eigenvectors, D is the diagonal matrix of eigenvalues.In the booklet	$M^{n} = P D^{n} P^{- 1}$	In the booklet

Functions

Lines, quadratics, exponential functions and polynomial roots. AI prints very little here.

Formula	Equation	In the exam
Equations of a straight lineAA and AI, SL and HL.In the booklet	$y = m x + c; a x + b y + d = 0; y - y_{1} = m (x - x_{1})$	In the booklet
Gradient from two pointsAA and AI, SL and HL.In the booklet	$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$	In the booklet
Axis of symmetry of a quadraticAA and AI, SL and HL. The vertex lies on this line.In the booklet	$f (x) = a x^{2} + b x + c \Rightarrow x = - \frac{b}{2 a}$	In the booklet
Quadratic formulaAA SL and HL; AI HL only (prior learning list); not printed for AI SL.In the booklet	$a x^{2} + b x + c = 0 \Rightarrow x = \frac{- b \pm b ^{2} - 4 a c}{2 a}, a \neq = 0$	In the booklet
DiscriminantAA SL and HL; AI HL only. Positive: two real roots; zero: one repeated root; negative: no real roots.In the booklet	$Δ = b^{2} - 4 a c$	In the booklet
Exponential and logarithmic functionsAA SL and HL only.In the booklet	$a^{x} = e^{x l n a}; lo g_{a} a^{x} = x = a^{l o g_{a} x}, a, x > 0, a \neq = 1$	In the booklet
Sum and product of roots of a polynomialAA HL only.In the booklet	$\sum_{r = 0}^{n} a_{r} x^{r} = 0 : sum = \frac{- a _{n - 1}}{a _{n}}, product = \frac{( - 1 ) ^{n} a _{0}}{a _{n}}$	In the booklet
Logistic functionAI HL only. L is the limiting value.In the booklet	$f (x) = \frac{L}{1 + C e ^{- k x}}, L, k, C > 0$	In the booklet

Geometry and trigonometry

Includes the prior learning list of areas and volumes, which both booklets print for all four courses.

Formula	Equation	In the exam
Area of a parallelogramPrior learning list. AA and AI, SL and HL.In the booklet	$A = bh$	In the booklet
Area of a trianglePrior learning list. AA and AI, SL and HL.In the booklet	$A = \frac{1}{2} bh$	In the booklet
Area of a trapezoidPrior learning list. AA and AI, SL and HL. a and b are the parallel sides.In the booklet	$A = \frac{1}{2} (a + b) h$	In the booklet
Area and circumference of a circlePrior learning list. AA and AI, SL and HL.In the booklet	$A = π r^{2}; C = 2 π r$	In the booklet
Volume of a cuboid, cylinder and prismPrior learning list. AA and AI, SL and HL. A is the cross-sectional area.In the booklet	$V = lw h; V = π r^{2} h; V = A h$	In the booklet
Curved surface area of a cylinderPrior learning list. AA and AI, SL and HL.In the booklet	$A = 2 π r h$	In the booklet
Distance between two points in two dimensionsPrior learning list. AA and AI, SL and HL.In the booklet	$d = (x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}$	In the booklet
Midpoint in two dimensionsIn the booklet	$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$	In the booklet
Distance between two points in three dimensionsAA and AI, SL and HL.In the booklet	$d = (x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} + (z_{1} - z_{2})^{2}$	In the booklet
Midpoint in three dimensionsIn the booklet	$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})$	In the booklet
Volume of a right pyramidAA and AI, SL and HL. A is the base area.In the booklet	$V = \frac{1}{3} A h$	In the booklet
Volume and curved surface area of a right coneAA and AI, SL and HL. l is the slant height.In the booklet	$V = \frac{1}{3} π r^{2} h; A = π r l$	In the booklet
Volume and surface area of a sphereAA and AI, SL and HL.In the booklet	$V = \frac{4}{3} π r^{3}; A = 4 π r^{2}$	In the booklet
Sine ruleAA and AI, SL and HL. Watch for the ambiguous case when given two sides and a non-included angle.In the booklet	$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$	In the booklet
Cosine ruleAA and AI, SL and HL.In the booklet	$c^{2} = a^{2} + b^{2} - 2 ab cos C; cos C = \frac{a ^{2} + b ^{2} - c ^{2}}{2 ab}$	In the booklet
Area of a triangle using sineAA and AI, SL and HL.In the booklet	$A = \frac{1}{2} ab sin C$	In the booklet
Arc length and sector area (radians)AA SL and HL; AI HL only. θ in radians.In the booklet	$l = r θ; A = \frac{1}{2} r^{2} θ$	In the booklet
Arc length and sector area (degrees)AI SL and HL only; not printed for AA. θ in degrees.In the booklet	$l = \frac{θ}{360} \times 2 π r; A = \frac{θ}{360} \times π r^{2}$	In the booklet
Tangent identityAA SL and HL; AI HL only.In the booklet	$tan θ = \frac{s i n θ}{c o s θ}$	In the booklet
Pythagorean identityAA SL and HL; AI HL only.In the booklet	$cos^{2} θ + sin^{2} θ = 1$	In the booklet
Double angle identities for sine and cosineAA SL and HL only.In the booklet	$sin 2 θ = 2 sin θ cos θ; cos 2 θ = cos^{2} θ - sin^{2} θ = 2 cos^{2} θ - 1 = 1 - 2 sin^{2} θ$	In the booklet
Reciprocal trigonometric identitiesAA HL only.In the booklet	$sec θ = \frac{1}{c o s θ}; cosec θ = \frac{1}{s i n θ}$	In the booklet
Further Pythagorean identitiesAA HL only.In the booklet	$1 + tan^{2} θ = sec^{2} θ; 1 + cot^{2} θ = cosec^{2} θ$	In the booklet
Compound angle identitiesAA HL only.In the booklet	$sin (A \pm B) = sin A cos B \pm cos A sin B; cos (A \pm B) = cos A cos B \mp sin A sin B; tan (A \pm B) = \frac{t a n A \pm t a n B}{1 \mp t a n A t a n B}$	In the booklet
Double angle identity for tangentAA HL only.In the booklet	$tan 2 θ = \frac{2 t a n θ}{1 - t a n ^{2} θ}$	In the booklet
Reflection matrix in the line y = (tan θ)xAI HL only.In the booklet	$(cos 2 θ sin 2 θ sin 2 θ - cos 2 θ)$	In the booklet
Stretch matricesAI HL only. k is the scale factor.In the booklet	$(k 0 01) (parallel to x -axis); (10 0 k) (parallel to y -axis)$	In the booklet
Enlargement matrixAI HL only. Scale factor k, centre (0, 0).In the booklet	$(k 0 0 k)$	In the booklet
Rotation matrices about the originAI HL only. θ > 0.In the booklet	$(cos θ sin θ - sin θ cos θ) (anticlockwise); (cos θ - sin θ sin θ cos θ) (clockwise)$	In the booklet
Magnitude of a vectorAA HL and AI HL only.In the booklet	$∣ v ∣ = v_{1}^{2} + v_{2}^{2} + v_{3}^{2}$	In the booklet
Scalar product and angle between vectorsAA HL and AI HL only.In the booklet	$v \cdot w = v_{1} w_{1} + v_{2} w_{2} + v_{3} w_{3} = ∣ v ∣∣ w ∣ cos θ; cos θ = \frac{v _{1} w _{1} + v _{2} w _{2} + v _{3} w _{3}}{∣ v ∣∣ w ∣}$	In the booklet
Vector equation of a lineAA HL and AI HL only. a is a point on the line, b the direction.In the booklet	$r = a + λ b$	In the booklet
Parametric form of a lineAA HL and AI HL only.In the booklet	$x = x_{0} + λ l, y = y_{0} + λm, z = z_{0} + λn$	In the booklet
Cartesian equations of a lineAA HL only.In the booklet	$\frac{x - x _{0}}{l} = \frac{y - y _{0}}{m} = \frac{z - z _{0}}{n}$	In the booklet
Vector product and area of a parallelogramAA HL and AI HL only. The area is for the parallelogram with v and w as adjacent sides.In the booklet	$v \times w = v_{2} w_{3} - v_{3} w_{2} v_{3} w_{1} - v_{1} w_{3} v_{1} w_{2} - v_{2} w_{1}; ∣ v \times w ∣ = ∣ v ∣∣ w ∣ sin θ; A = ∣ v \times w ∣$	In the booklet
Vector equation of a planeAA HL only.In the booklet	$r = a + λ b + μ c$	In the booklet
Equation of a plane using the normal vectorAA HL only.In the booklet	$r \cdot n = a \cdot n$	In the booklet
Cartesian equation of a planeAA HL only.In the booklet	$a x + b y + cz = d$	In the booklet

Statistics and probability

AI SL prints no standard deviation or normal formula, so use the calculator for those.

Formula	Equation	In the exam
Interquartile rangeAA and AI, SL and HL.In the booklet	$IQR = Q_{3} - Q_{1}$	In the booklet
Mean of grouped or frequency dataAA and AI, SL and HL.In the booklet	$\overset{x}{ˉ} = \frac{\sum _{i = 1}^{k} f _{i} x _{i}}{n}, n = \sum_{i = 1}^{k} f_{i}$	In the booklet
Probability of an event and complementAA and AI, SL and HL.In the booklet	$P (A) = \frac{n ( A )}{n ( U )}; P (A) + P (A^{'}) = 1$	In the booklet
Combined eventsAA and AI, SL and HL.In the booklet	$P (A \cup B) = P (A) + P (B) - P (A \cap B)$	In the booklet
Mutually exclusive eventsAA and AI, SL and HL. Only when A and B cannot both happen.In the booklet	$P (A \cup B) = P (A) + P (B)$	In the booklet
Conditional probabilityAA and AI, SL and HL.In the booklet	$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$	In the booklet
Independent eventsAA and AI, SL and HL.In the booklet	$P (A \cap B) = P (A) P (B)$	In the booklet
Expected value of a discrete random variableAA and AI, SL and HL.In the booklet	$E (X) = \sum x P (X = x)$	In the booklet
Binomial distribution: mean and varianceAA and AI, SL and HL. The probability formula itself is not printed.In the booklet	$X \sim B (n, p) : E (X) = n p, Var (X) = n p (1 - p)$	In the booklet
Standardised normal variableAA SL and HL only; not printed for AI.In the booklet	$z = \frac{x - μ}{σ}$	In the booklet
Bayes' theoremAA HL only. The booklet also prints the version with three events B₁, B₂, B₃.In the booklet	$P (B ∣ A) = \frac{P ( B ) P ( A ∣ B )}{P ( B ) P ( A ∣ B ) + P ( B ^{'} ) P ( A ∣ B ^{'} )}$	In the booklet
Variance and standard deviation of grouped dataAA HL only.In the booklet	$σ^{2} = \frac{\sum f _{i} ( x _{i} - μ ) ^{2}}{n} = \frac{\sum f _{i} x _{i}^{2}}{n} - μ^{2}; σ = \frac{\sum f _{i} ( x _{i} - μ ) ^{2}}{n}$	In the booklet
Linear transformation of a random variableAA HL and AI HL only. The constant b does not change the variance.In the booklet	$E (a X + b) = a E (X) + b; Var (a X + b) = a^{2} Var (X)$	In the booklet
Expected value and variance of a continuous random variableAA HL only.In the booklet	$E (X) = μ = \int_{- \infty}^{\infty} x f (x) d x; Var (X) = E (X^{2}) - [E (X)]^{2} = \int_{- \infty}^{\infty} x^{2} f (x) d x - μ^{2}$	In the booklet
Variance of a discrete random variableAA HL only.In the booklet	$Var (X) = \sum (x - μ)^{2} P (X = x) = \sum x^{2} P (X = x) - μ^{2}$	In the booklet
Linear combinations of independent random variablesAI HL only. Variances always add, even when the variables are subtracted.In the booklet	$E (a_{1} X_{1} \pm \dots \pm a_{n} X_{n}) = a_{1} E (X_{1}) \pm \dots \pm a_{n} E (X_{n}); Var (a_{1} X_{1} \pm \dots \pm a_{n} X_{n}) = a_{1}^{2} Var (X_{1}) + \dots + a_{n}^{2} Var (X_{n})$	In the booklet
Unbiased estimate of population varianceAI HL only.In the booklet	$s_{n - 1}^{2} = \frac{n}{n - 1} s_{n}^{2}$	In the booklet
Poisson distribution: mean and varianceAI HL only.In the booklet	$X \sim Po (m) : E (X) = m, Var (X) = m$	In the booklet
Transition matricesAI HL only. s₀ is the initial state.In the booklet	$T^{n} s_{0} = s_{n}$	In the booklet

Calculus

AA SL prints the core rules; AI SL prints only the power rule, the area under a curve and the trapezoidal rule.

Formula	Equation	In the exam
Derivative of x to the power nAA and AI, SL and HL.In the booklet	$f (x) = x^{n} \Rightarrow f^{'} (x) = n x^{n - 1}$	In the booklet
Integral of x to the power nAA and AI, SL and HL.In the booklet	$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C, n \neq = - 1$	In the booklet
Area under a curveAA and AI, SL and HL. The curve must be above the x-axis.In the booklet	$A = \int_{a}^{b} y d x, f (x) > 0$	In the booklet
Trapezoidal ruleAI SL and HL only; not printed for AA.In the booklet	$\int_{a}^{b} y d x \approx \frac{1}{2} h ((y_{0} + y_{n}) + 2 (y_{1} + y_{2} + \dots + y_{n - 1})), h = \frac{b - a}{n}$	In the booklet
Derivatives of sin x, cos x, e^x and ln xAA SL and HL; AI HL only.In the booklet	$sin x \to cos x; cos x \to - sin x; e^{x} \to e^{x}; ln x \to \frac{1}{x}$	In the booklet
Chain ruleAA SL and HL; AI HL only.In the booklet	$y = g (u), u = f (x) \Rightarrow \frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$	In the booklet
Product ruleAA SL and HL; AI HL only.In the booklet	$y = uv \Rightarrow \frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}$	In the booklet
Quotient ruleAA SL and HL; AI HL only.In the booklet	$y = \frac{u}{v} \Rightarrow \frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v ^{2}}$	In the booklet
Derivative of tan xAI HL only. AA HL prints the same result as sec squared x in its HL derivatives list.In the booklet	$tan x \to \frac{1}{c o s ^{2} x}$	In the booklet
Acceleration, distance travelled and displacementAA SL and HL; AI HL only. AI HL also prints a = v dv/ds. Distance uses the modulus of v, so split the integral where v changes sign.In the booklet	$a = \frac{d v}{d t} = \frac{d ^{2} s}{d t ^{2}}; distance = \int_{t_{1}}^{t_{2}} ∣ v (t) ∣ d t; displacement = \int_{t_{1}}^{t_{2}} v (t) d t$	In the booklet
Standard integrals (1/x, sin x, cos x, e^x)AA SL and HL; AI HL only. AI HL also prints the integral of 1 over cos squared x, which is tan x plus C.In the booklet	$\int \frac{1}{x} d x = ln ∣ x ∣ + C; \int sin x d x = - cos x + C; \int cos x d x = sin x + C; \int e^{x} d x = e^{x} + C$	In the booklet
Area between a curve and the y-axisAA HL and AI HL only.In the booklet	$A = \int_{a}^{b} x d y$	In the booklet
Volume of revolutionAA HL and AI HL only.In the booklet	$V = \int_{a}^{b} π y^{2} d x or V = \int_{a}^{b} π x^{2} d y$	In the booklet
Euler's methodAA HL and AI HL only. h is the step length.In the booklet	$y_{n + 1} = y_{n} + h \times f (x_{n}, y_{n}); x_{n + 1} = x_{n} + h$	In the booklet
Euler's method for coupled systemsAI HL only.In the booklet	$x_{n + 1} = x_{n} + h f_{1} (x_{n}, y_{n}, t_{n}); y_{n + 1} = y_{n} + h f_{2} (x_{n}, y_{n}, t_{n}); t_{n + 1} = t_{n} + h$	In the booklet
Exact solution of coupled linear differential equationsAI HL only. λ are eigenvalues, p are eigenvectors.In the booklet	$x = A e^{λ_{1} t} p_{1} + B e^{λ_{2} t} p_{2}$	In the booklet
Derivative from first principlesAA HL only.In the booklet	$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$	In the booklet
Derivatives of tan, sec, cosec and cotAA HL only.In the booklet	$tan x \to sec^{2} x; sec x \to sec x tan x; cosec x \to - cosec x cot x; cot x \to - cosec^{2} x$	In the booklet
Derivatives of a^x, log_a x and inverse trigonometric functionsAA HL only.In the booklet	$a^{x} \to a^{x} ln a; lo g_{a} x \to \frac{1}{x l n a}; arcsin x \to \frac{1}{1 - x ^{2}}; arccos x \to \frac{- 1}{1 - x ^{2}}; arctan x \to \frac{1}{1 + x ^{2}}$	In the booklet
Further standard integralsAA HL only.In the booklet	$\int a^{x} d x = \frac{1}{l n a} a^{x} + C; \int \frac{d x}{a ^{2} + x ^{2}} = \frac{1}{a} arctan \frac{x}{a} + C; \int \frac{d x}{a ^{2} - x ^{2}} = arcsin \frac{x}{a} + C, ∣ x ∣ < a$	In the booklet
Integration by partsAA HL only.In the booklet	$\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x or \int u d v = uv - \int v d u$	In the booklet
Integrating factorAA HL only.In the booklet	$For y^{'} + P (x) y = Q (x) : e^{\int P (x) d x}$	In the booklet
Maclaurin seriesAA HL only.In the booklet	$f (x) = f (0) + x f^{'} (0) + \frac{x ^{2}}{2 !} f^{''} (0) + \dots$	In the booklet
Maclaurin series of standard functionsAA HL only.In the booklet	$e^{x} = 1 + x + \frac{x ^{2}}{2 !} + \dots; ln (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots; sin x = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots; cos x = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots; arctan x = x - \frac{x ^{3}}{3} + \frac{x ^{5}}{5} - \dots$	In the booklet

Learn these: not in the booklet

Laws of exponents (AA and AI) $a^{m} \times a^{n} = a^{m + n}; \frac{a ^{m}}{a ^{n}} = a^{m - n}; (a^{m})^{n} = a^{mn}; a^{- n} = \frac{1}{a ^{n}}; a^{1/ n} = n a$
Special logarithm values (AA, and AI HL) $lo g_{a} 1 = 0; lo g_{a} a = 1$
Parallel and perpendicular gradients (AA and AI) $m_{1} = m_{2} (parallel); m_{1} m_{2} = - 1 (perpendicular)$
Other forms of a quadratic (AA and AI) $f (x) = a (x - p) (x - q) gives roots p, q; f (x) = a (x - h)^{2} + k gives vertex (h, k)$
Percentage change (AA and AI) $\frac{new - old}{old} \times 100%$
Sinusoidal model (AA and AI) $f (x) = a sin (b (x - c)) + d; amplitude ∣ a ∣; period \frac{2 π}{b} or \frac{36 0 ^{\circ}}{b}; midline y = d$
Right-angled trigonometry (AA and AI) $sin θ = \frac{opp}{hyp}; cos θ = \frac{adj}{hyp}; tan θ = \frac{opp}{adj}; a^{2} + b^{2} = c^{2}$
Exact values (AA) $sin 3 0^{\circ} = \frac{1}{2}; sin 4 5^{\circ} = \frac{2}{2}; sin 6 0^{\circ} = \frac{3}{2}; tan 3 0^{\circ} = \frac{3}{3}; tan 4 5^{\circ} = 1; tan 6 0^{\circ} = 3$
Radians and degrees (AA and AI HL) $π rad = 18 0^{\circ}$
Trigonometric symmetries (AA) $sin (π - θ) = sin θ; cos (π - θ) = - cos θ; sin (\frac{π}{2} - θ) = cos θ; cos (\frac{π}{2} - θ) = sin θ; sin (- θ) = - sin θ; cos (- θ) = cos θ$
Modulus and conjugate of a complex number (AA HL and AI HL) $∣ z ∣ = a^{2} + b^{2}; z^{*} = a - bi; z z^{*} = ∣ z ∣^{2}$
Factor and remainder theorems (AA HL) $f (a) = 0 \Rightarrow (x - a) is a factor; remainder on dividing by (x - a) is f (a)$
Vector basics (AA HL and AI HL) $A B = b - a; v \cdot w = 0 \Rightarrow perpendicular; area of triangle = \frac{1}{2} ∣ v \times w ∣$
Outliers (AA and AI) $x < Q_{1} - 1.5 \times IQR or x > Q_{3} + 1.5 \times IQR$
Binomial probability (AA and AI) $P (X = r) = (r n) p^{r} (1 - p)^{n - r}$
Standardising for a normal distribution (AI; printed for AA only) $z = \frac{x - μ}{σ}$
Chi-squared test (AI) $χ^{2} = \sum \frac{( f _{o} - f _{e} ) ^{2}}{f _{e}}; f_{e} = \frac{row total \times column total}{grand total}; degrees of freedom = (r - 1) (c - 1)$
Poisson probability (AI HL) $P (X = r) = \frac{e ^{- m} m ^{r}}{r !}$
Distribution of the sample mean (AI HL) $E (\overset{ˉ}{X}) = μ; Var (\overset{ˉ}{X}) = \frac{σ ^{2}}{n}$
Tangent and normal at x = a (AA and AI) $tangent y - f (a) = f^{'} (a) (x - a); normal gradient = - \frac{1}{f ^{'} ( a )}$
Stationary points and curvature (AA and AI) $f^{'} (x) = 0 at a stationary point; f^{''} (x) > 0 minimum; f^{''} (x) < 0 maximum; f^{''} (x) = 0 with a sign change: point of inflection$
Velocity (AA and AI HL) $v = \frac{d s}{d t}$
Reverse chain rule (AA) $\int f (a x + b) d x = \frac{1}{a} F (a x + b) + C$
Area between two curves (AA) $A = \int_{a}^{b} (f (x) - g (x)) d x where f (x) \geq g (x)$
Integration by substitution (AA HL) $\int f (g (x)) g^{'} (x) d x = \int f (u) d u$
L'Hôpital's rule (AA HL) $lim_{x \to a} \frac{f ( x )}{g ( x )} = lim_{x \to a} \frac{f ^{'} ( x )}{g ^{'} ( x )} for \frac{0}{0} or \frac{\infty}{\infty}$
Separable differential equations (AA HL) $\frac{d y}{d x} = f (x) g (y) \Rightarrow \int \frac{1}{g ( y )} d y = \int f (x) d x$
Homogeneous differential equations (AA HL) $substitute y = vx, \frac{d y}{d x} = v + x \frac{d v}{d x}$
Eigenvalues and eigenvectors (AI HL) $M v = λ v; det (M - λ I) = 0$
Steady state of a transition matrix (AI HL) $T s = s$

Study booklets

Our printable version of each board's official booklet: everything it prints (formulas, constants, data and statistical tables) in the board's own order, plus what you must learn and exam tips. Open the dropdown, then click a booklet to open it or use Download to save the PDF.

IB Diploma2 booklets

Maths: analysis and approaches SL and HLPDF, 9 pages, 0.6 MB
Download
Maths: applications and interpretation SL and HLPDF, 8 pages, 0.5 MB
Download

How to use the formula sheet in the exam

Open the booklet in practice sessions, so you know where each topic sits and can find a row in seconds.
Check the course tag before relying on a row: AI SL students do not get the quadratic formula, log laws or trig identities printed.
Copy a printed formula carefully: the AA booklet version 1.0 shows the infinite geometric sum with r minus 1 on the bottom, but the correct denominator is 1 minus r.
Use the booklet for the pattern, then check units: compound interest needs the rate as r%, so r = 5 means 5 per cent, not 0.05.
For anything the booklet does not print, such as tangent and normal equations, write the rule down before substituting.

Common questions

Is the formula booklet given in the IB maths exam?

The booklet's cover says it is for use during the course and in the examinations. Your school should supply a copy for the exam, and you should practise with it all year.

Is the AA formula booklet the same for SL and HL?

Yes, AA SL and AA HL share one booklet, and the contents page splits each topic into SL and HL sections. AI SL and AI HL likewise share one booklet, separate from AA.

Does the AI booklet print the quadratic formula?

Only in its HL part of the prior learning section. The AA booklet prints it for both SL and HL. AI SL students usually solve quadratics with the calculator, but learning the formula is still sensible.

Why is my row marked AA only or AI only?

The two courses teach different content: AA prints more trigonometric identities, vector geometry and calculus, while AI includes matrices, Poisson, transition matrices, differential systems and the trapezoidal rule. Each row names the booklets that print it.

What do I need to memorise that is not in the booklet?

Use the recall list on this page. It covers results such as laws of exponents, exact trigonometric values, outlier rules and tangent equations. Confirm the exact syllabus wording in your course guide with your teacher.

Practise this topic

Real past paper questions, by topic

CuratedPrep worksheets group real past paper questions by syllabus topic, with a full marking scheme and revision notes, for every IB maths topic, SL and HL.

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Formulas checked against the official documents linked above. Boards update their booklets from time to time: always practise with the version your exam uses.

IB maths formula booklet guide: AA and AI, SL and HL

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