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IB Math AA HL Formula Booklet

A searchable IB Math AA HL formula booklet with core and higher-level formulas, definitions, annotations, and clear KaTeX typesetting.

These formulas are rendered as real mathematics rather than images. Read the definition, check every variable, and follow the topic link to see the idea in context. Always confirm the permitted official booklet or sheet for your examination session.

Reference section

A · Number & Algebra

Arithmetic sequences & series

u_n=u_1+(n-1)d \qquad S_n=\frac{n}{2}(2u_1+(n-1)d)=\frac{n}{2}(u_1+u_n)

Common difference d = u_2 - u_1. Use the S_n form with u_n directly when the last term is already known — it's usually faster.

$u_n$: nth term
$u_1$: first term
$d$: common difference
$n$: number of terms
$S_n$: sum of the first n terms

Geometric sequences & series

u_n=u_1r^{n-1} \qquad S_n=\frac{u_1(r^n-1)}{r-1}=\frac{u_1(1-r^n)}{1-r}

Common ratio r = u_2/u_1. Use whichever form of S_n avoids a negative denominator.

$u_n$: nth term
$u_1$: first term
$r$: common ratio
$n$: number of terms
$S_n$: sum of the first n terms

Sum to infinity

S_\infty=\frac{u_1}{1-r} \qquad (|r|<1)

Only exists when the common ratio has magnitude strictly less than 1 — otherwise the series diverges.

$S_\infty$: sum to infinity
$u_1$: first term
$r$: common ratio, -1<r<1

Compound interest

FV=PV\left(1+\frac{i}{100}\right)^n

If interest is compounded more than once per year, divide i by the number of periods per year and multiply n by that same number.

$FV$: future value
$PV$: present value
$i$: interest rate per compounding period (%)
$n$: number of compounding periods

Binomial theorem

(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r \qquad \binom{n}{r}=\frac{n!}{r!(n-r)!}

a is raised to the power (n−r), b to the power r. The general term is useful for finding one specific coefficient without a full expansion.

$n$: the power
$r$: term index, 0 to n
$\binom{n}{r}$: binomial coefficient (also written nCr)

Counting principles

{}^nP_r=\frac{n!}{(n-r)!} \qquad {}^nC_r=\binom{n}{r}=\frac{n!}{r!(n-r)!}

Permutations (nPr) count arrangements where order matters; combinations (nCr) count selections where it does not.

$n$: total number of distinct objects
$r$: number chosen/arranged

Extended binomial series

(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\ldots \quad (|x|<1)

Valid for any real n (not just positive integers), but only converges for |x|<1. Manipulate expressions into the form (1+x)^n first if needed.

$n$: any real exponent
$x$: requires |x|<1 for convergence

Complex numbers: polar/Euler form

z=r(\cos\theta+i\sin\theta)=re^{i\theta} \qquad |z|=r=\sqrt{a^2+b^2}

Converts Cartesian a+bi to polar/Euler form. Multiplying multiplies moduli and adds arguments; dividing divides moduli and subtracts arguments.

$z$: a complex number
$r$: modulus, |z|
$\theta$: argument, arg(z)
$a,b$: real and imaginary parts

De Moivre's theorem & nth roots

[r(\cos\theta+i\sin\theta)]^n=r^n(\cos n\theta+i\sin n\theta) \qquad z^{1/n}=r^{1/n}\,\text{cis}\left(\frac{\theta+2k\pi}{n}\right)

Every non-zero complex number has exactly n distinct nth roots, evenly spaced by angle 2π/n, for k = 0, 1, ..., n-1.

$r$: modulus
$\theta$: argument
$n$: the power (or root)
$k$: root index, 0 to n-1

Sum & product of polynomial roots

\alpha+\beta+\gamma=-\frac{b}{a} \qquad \alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a} \qquad \alpha\beta\gamma=-\frac{d}{a}

For a cubic ax³+bx²+cx+d=0 with roots α, β, γ. Signs alternate with each successive symmetric function.

$a,b,c,d$: the cubic's coefficients
$\alpha,\beta,\gamma$: the three roots

Reference section

B · Functions

Quadratic formula & discriminant

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \qquad \Delta=b^2-4ac

Δ>0: two distinct real roots. Δ=0: one repeated root. Δ<0: no real roots.

$a,b,c$: coefficients of ax²+bx+c=0
$\Delta$: the discriminant

Vertex of a quadratic

h=-\frac{b}{2a} \qquad k=f(h)

Gives the turning point (h,k) of y=ax²+bx+c directly from the general form.

$h,k$: coordinates of the vertex
$a,b$: coefficients of the quadratic

Composite & inverse functions

(f\circ g)(x)=f(g(x)) \qquad f(f^{-1}(x))=f^{-1}(f(x))=x

The graph of f⁻¹ is the reflection of f's graph in the line y=x. An inverse only exists over the whole domain if f is one-to-one.

$f\circ g$: f composed with g — apply g first
$f^{-1}$: the inverse function of f

Rational function asymptotes

y=\frac{ax+b}{cx+d}: \quad x=-\frac{d}{c}\ (\text{vertical}), \quad y=\frac{a}{c}\ (\text{horizontal})

Vertical asymptote where the denominator is zero; horizontal asymptote from the ratio of leading coefficients.

$a,b,c,d$: the rational function's coefficients

Reference section

C · Geometry & Trigonometry

Sine rule & cosine rule

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \qquad a^2=b^2+c^2-2bc\cos A

Sine rule: use with an angle-side opposite pair. Cosine rule: use with SSS or SAS. Watch for the ambiguous case (SSA) with the sine rule.

$a,b,c$: sides, opposite angles A, B, C respectively

Area of a triangle

\text{Area}=\frac12ab\sin C

C must be the angle included between the two given sides a and b.

$a,b$: two sides
$C$: the angle between them

Radians, arc length & sector area

l=r\theta \qquad A=\frac12r^2\theta

Both formulas require θ in radians, not degrees. One radian is the angle where arc length equals the radius.

$l$: arc length
$A$: sector area
$r$: radius
$\theta$: angle in radians

Pythagorean identity

\sin^2\theta+\cos^2\theta=1 \qquad \tan\theta=\frac{\sin\theta}{\cos\theta}

Follows from the unit circle definition. Rearranges to sin²θ=1−cos²θ or cos²θ=1−sin²θ.

$\theta$: any angle

Compound & double angle identities

\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B \qquad \cos(2\theta)=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta

The three forms of cos(2θ) are equivalent via the Pythagorean identity — choose whichever is convenient.

$A,B$: two angles
$\theta$: a single angle, for the double-angle forms

Reciprocal trig functions

\sec\theta=\frac{1}{\cos\theta} \qquad \csc\theta=\frac{1}{\sin\theta} \qquad \cot\theta=\frac{1}{\tan\theta} \qquad 1+\tan^2\theta=\sec^2\theta

sec pairs with cos, csc pairs with sin — the letter association is easy to swap by mistake.

$\theta$: any angle where the function is defined

Vectors: magnitude, scalar & vector product

|\vec{v}|=\sqrt{v_1^2+v_2^2+v_3^2} \qquad \vec{u}\cdot\vec{v}=|\vec{u}||\vec{v}|\cos\theta \qquad |\vec{u}\times\vec{v}|=|\vec{u}||\vec{v}|\sin\theta

The scalar product returns a number (zero exactly when perpendicular); the vector product returns a vector perpendicular to both, with magnitude equal to the area of the parallelogram they span.

$\vec{u},\vec{v}$: vectors
$\theta$: angle between them

Vector equation of a line

\vec{r}=\vec{a}+t\vec{d}

a is a known point on the line, d is a direction vector (not unique — any scalar multiple works), t is a scalar parameter.

$\vec{r}$: position vector of a general point on the line
$\vec{a}$: a known point
$\vec{d}$: direction vector
$t$: parameter

Equation of a plane

\vec{n}\cdot(\vec{r}-\vec{a})=0 \quad \text{or} \quad n_1x+n_2y+n_3z=d

n is a normal vector to the plane, found via the cross product of two direction vectors within the plane.

$\vec{n}$: normal vector
$\vec{a}$: a known point on the plane
$d$: constant, found from n · a (the dot product of the normal vector and a known point)

Reference section

D · Statistics & Probability

Mean, variance & standard deviation

\bar{x}=\frac{\sum f_ix_i}{\sum f_i} \qquad \sigma^2=\frac{\sum(x_i-\bar{x})^2}{n} \qquad \sigma=\sqrt{\sigma^2}

For frequency/grouped data, x_i is the value (or class midpoint) and f_i its frequency.

$\bar{x}$: mean
$\sigma$: standard deviation
$x_i$: each data value
$f_i$: frequency of x_i
$n$: number of data points

Effect of a linear transformation

y=ax+b: \quad \bar{y}=a\bar{x}+b \qquad \sigma_y=|a|\sigma_x

A constant shift (b) changes the mean but not the spread. A scale factor (a) changes both — the standard deviation scales by |a|.

$a,b$: transformation constants
$\bar{x},\sigma_x$: original mean and standard deviation

Probability rules

P(A')=1-P(A) \qquad P(A\cup B)=P(A)+P(B)-P(A\cap B) \qquad P(A|B)=\frac{P(A\cap B)}{P(B)}

For mutually exclusive events, P(A∩B)=0. For independent events, P(A∩B)=P(A)P(B) and P(A|B)=P(A).

$A,B$: events
$A'$: complement of A (not A)

Discrete random variable: expected value

E(X)=\sum x_iP(X=x_i)

The long-run average outcome over many repetitions — need not be a value X can actually take.

$E(X)$: expected value
$x_i$: possible values of X
$P(X=x_i)$: probability of each value

Binomial distribution

X\sim B(n,p): \quad P(X=k)=\binom{n}{k}p^k(1-p)^{n-k} \qquad E(X)=np \qquad \text{Var}(X)=np(1-p)

Requires: fixed number of trials n, independent trials, two outcomes per trial, constant success probability p.

$n$: number of trials
$p$: probability of success on each trial
$k$: number of successes

Normal distribution: standardization

z=\frac{x-\mu}{\sigma}

Converts any normal distribution to the standard normal (mean 0, standard deviation 1) for comparison and probability lookup.

$z$: standardized score
$x$: raw value
$\mu$: population mean
$\sigma$: population standard deviation

Bayes' theorem

P(A|B)=\frac{P(B|A)P(A)}{P(B)} \qquad P(B)=P(B|A)P(A)+P(B|A')P(A')

Reverses a conditional probability — use a tree diagram to organise the known probabilities before applying the formula.

$A,B$: events
$A'$: complement of A

Linear combinations of random variables

E(aX+b)=aE(X)+b \qquad \text{Var}(aX+b)=a^2\text{Var}(X) \qquad \text{Var}(X\pm Y)=\text{Var}(X)+\text{Var}(Y)\ (X,Y\text{ independent})

Variance scales by a², not a. Variances add for both sums AND differences of independent variables — never subtract them.

$a,b$: constants
$X,Y$: independent random variables

Reference section

E · Calculus

Derivative from first principles

f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}

The formal definition of the derivative — the limiting gradient of a secant line as the interval shrinks to zero.

$h$: a small change in x, approaching 0

Standard derivatives

\frac{d}{dx}(x^n)=nx^{n-1} \qquad \frac{d}{dx}(\sin x)=\cos x \qquad \frac{d}{dx}(\cos x)=-\sin x \qquad \frac{d}{dx}(e^x)=e^x \qquad \frac{d}{dx}(\ln x)=\frac1x

Differentiation is linear — differentiate sums term by term, keeping constant coefficients.

$n$: any real exponent, for the power rule

Chain, product & quotient rules

\frac{d}{dx}f(g(x))=f'(g(x))g'(x) \qquad (fg)'=f'g+fg' \qquad \left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}

Chain rule: differentiate outside, keep inside, times derivative of inside. Quotient rule order matters: f′g minus fg′, over g squared.

$f,g$: functions of x

Standard integrals

\int x^n\,dx=\frac{x^{n+1}}{n+1}+c\ (n\neq-1) \qquad \int\cos x\,dx=\sin x+c \qquad \int\sin x\,dx=-\cos x+c \qquad \int e^x\,dx=e^x+c \qquad \int\frac1x\,dx=\ln|x|+c

Always include the constant of integration c for an indefinite integral.

$c$: constant of integration

Definite integral & area

\int_a^b f(x)\,dx=F(b)-F(a) \quad \text{where } F\prime(x)=f(x)

Gives signed area — split at any root of f(x) within [a,b] and take absolute values to find total (unsigned) area.

$a,b$: limits of integration
$F$: an antiderivative of f

Trapezoidal rule

\int_a^b f(x)\,dx\approx\frac{h}{2}\left[y_0+2(y_1+\ldots+y_{n-1})+y_n\right]

Overestimates area for concave-up curves, underestimates for concave-down curves. All interior y-values are doubled.

$h$: strip width, (b-a)/n
$n$: number of strips
$y_0,\ldots,y_n$: function values at each x

Kinematics via calculus

v(t)=s\prime(t) \qquad a(t)=v\prime(t)=s\prime\prime(t)

Object at rest when v(t)=0. Total distance (unlike displacement) requires splitting the motion at every point where v=0.

$s(t)$: displacement
$v(t)$: velocity
$a(t)$: acceleration

Derivatives of further functions

\frac{d}{dx}(\sec x)=\sec x\tan x \qquad \frac{d}{dx}(\arcsin x)=\frac{1}{\sqrt{1-x^2}} \qquad \frac{d}{dx}(\arctan x)=\frac{1}{1+x^2}

Combine with the chain rule for composite versions, e.g. d/dx[arctan(kx)] = k/(1+k²x²).

$x$: the variable

Integration by parts

\int u\,dv=uv-\int v\,du

Choose u to simplify on differentiation (LIATE order: Log, Inverse trig, Algebraic, Trig, Exponential); dv should be easy to integrate.

$u,v$: the two chosen parts of the product

Volume of revolution

V=\pi\int_a^b[f(x)]^2\,dx

Disc method for rotation about the x-axis. Do not forget to square f(x) before integrating.

$f(x)$: the function being rotated
$a,b$: limits of integration

Differential equations & Euler's method

\int\frac{1}{g(y)}\,dy=\int f(x)\,dx \quad \text{(separable ODE)} \qquad y_{n+1}=y_n+h\cdot f(x_n,y_n)\ \text{(Euler's method)}

Separation of variables solves dy/dx=f(x)g(y) exactly. Euler's method numerically approximates a solution using small steps h.

$h$: Euler step size
$f(x_n,y_n)$: gradient at the current approximated point

Maclaurin series

f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\ldots

Standard series worth memorising: e^x, sin x, cos x (see topic notes). Most accurate near x=0.

$f^{(n)}(0)$: the nth derivative of f evaluated at x=0

Reviewed by the Study to Learn editorial team · Updated 2026-07-24