HL AA Formula Booklet — Analysis & Approaches HL

All 24 formulas you need for IB Mathematics Analysis & Approaches HL, grouped by syllabus topic and written in the same notation as the official IB formula booklet. Use it to revise which formulas are given in the exam, and which you still need to understand how to apply.

Open the print-ready one-page booklet → Practise HL AA questions
Everything in SL AA, plus… · Topic 1 (HL only) — Algebra extension · Topic 3 (HL only) — Geometry & Vectors · Topic 4 (HL only) — Statistics & Probability extension · Topic 5 (HL only) — Calculus extension

Everything in SL AA, plus…

Complex numbers — polar (Cartesian → polar)
\(z = a + bi = r\,\mathrm{cis}\,\theta = re^{i\theta}\)
Modulus & argument
\(r = |z| = \sqrt{a^2 + b^2},\ \theta = \arg z\)
Multiplication / division in polar form
\(z_1 z_2 = r_1 r_2 \,\mathrm{cis}(\theta_1 + \theta_2),\ \tfrac{z_1}{z_2} = \tfrac{r_1}{r_2}\,\mathrm{cis}(\theta_1 - \theta_2)\)
De Moivre's theorem
\(\bigl(r\,\mathrm{cis}\,\theta\bigr)^n = r^n\,\mathrm{cis}(n\theta)\)
Euler's identity
\(e^{i\pi} + 1 = 0\)

Topic 1 (HL only) — Algebra extension

Sum of an infinite geometric series
\(S_\infty = \dfrac{u_1}{1 - r},\ |r| < 1\)
Proof by induction (skeleton)
\(P(1)\text{ true};\ P(k) \Rightarrow P(k+1);\ \therefore P(n)\text{ true for all }n\in\mathbb{Z}^+\)

Topic 3 (HL only) — Geometry & Vectors

Vector magnitude
\(|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\)
Dot product
\(\vec{a}\cdot\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\vec{a}||\vec{b}|\cos\theta\)
Cross product (magnitude)
\(|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta\)
Cross product (component)
\(\vec{a}\times\vec{b} = (a_2 b_3 - a_3 b_2,\, a_3 b_1 - a_1 b_3,\, a_1 b_2 - a_2 b_1)\)
Vector equation of a line
\(\vec{r} = \vec{a} + t\,\vec{d}\)
Cartesian equation of a plane
\(\vec{r}\cdot\vec{n} = \vec{a}\cdot\vec{n}\)
Compound-angle identities
\(\sin(A\pm B) = \sin A\cos B \pm \cos A\sin B\)
Compound-angle (cosine)
\(\cos(A\pm B) = \cos A\cos B \mp \sin A\sin B\)

Topic 4 (HL only) — Statistics & Probability extension

Bayes' theorem
\(P(A \mid B) = \dfrac{P(B \mid A) P(A)}{P(B)}\)
Variance (continuous)
\(\mathrm{Var}(X) = E(X^2) - \bigl[E(X)\bigr]^2\)
Linearity of expectation
\(E(aX + b) = aE(X) + b,\ \mathrm{Var}(aX + b) = a^2 \mathrm{Var}(X)\)

Topic 5 (HL only) — Calculus extension

Integration by parts
\(\int u\, dv = uv - \int v\, du\)
Standard integrals (HL)
\(\int \dfrac{1}{\sqrt{a^2-x^2}}\, dx = \arcsin\!\tfrac{x}{a} + C,\ \int \dfrac{1}{a^2 + x^2}\, dx = \tfrac{1}{a}\arctan\!\tfrac{x}{a} + C\)
Maclaurin series
\(f(x) = f(0) + f'(0)x + \dfrac{f''(0)}{2!}x^2 + \dfrac{f'''(0)}{3!}x^3 + \cdots\)
Standard Maclaurin expansions
\(e^x = \sum_{k=0}^{\infty}\tfrac{x^k}{k!},\ \sin x = \sum_{k=0}^{\infty}\tfrac{(-1)^k x^{2k+1}}{(2k+1)!},\ \cos x = \sum_{k=0}^{\infty}\tfrac{(-1)^k x^{2k}}{(2k)!}\)
Volume of revolution (about x-axis)
\(V = \pi \int_a^b y^2\, dx\)
Volume of revolution (about y-axis)
\(V = \pi \int_c^d x^2\, dy\)

Other courses: SL AI formulas · SL AA formulas · HL AI formulas · HL AA revision notes · HL AA questions by topic