Knowledge organisers

Pure Mathematics 3 (P3): Edexcel IAL Maths knowledge organiser

Everything to know about pure mathematics 3 (p3) on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Composite function: fg(x) = f(g(x)): apply g first.
Modulus function: |x| is the size of x without its sign; the graph of y = |f(x)| reflects the negative parts up.
R form: a sin θ + b cos θ = R sin(θ + α) with R = √(a² + b²).
Quotient rule: d/dx(u/v) = (v du/dx − u dv/dx)/v².

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Exponential formIn the formula booklet	\(e^{x\ln a}=a^x\)
Reciprocal identities	\(\sec^2\theta=1+\tan^2\theta,\) \(\cosec^2\theta=1+\cot^2\theta\)
Compound sinIn the formula booklet	\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\)
Compound cosIn the formula booklet	\(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\)
Compound tanIn the formula booklet	\(\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)
Double angle	\(\sin2A=2\sin A\cos A,\) \(\cos2A=2\cos^2A-1=1-2\sin^2A,\) \(\tan2A=\frac{2\tan A}{1-\tan^2A}\)
\(R\) form	\(a\sin\theta\pm b\cos\theta=R\sin(\theta\pm\alpha),\) \(R=\sqrt{a^2+b^2},\) \(\tan\alpha=\tfrac ba\)
Standard derivatives	\((e^{kx})'=ke^{kx},\) \((\ln x)'=\tfrac1x,\) \((\sin kx)'=k\cos kx,\) \((\cos kx)'=-k\sin kx\)
Trig derivativesIn the formula booklet	\((\tan kx)'=k\sec^2kx,\) \((\sec x)'=\sec x\tan x,\) \((\cot x)'=-\cosec^2x,\) \((\cosec x)'=-\cosec x\cot x\)
Product; chain	\((uv)'=uv'+vu';\) \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\)
QuotientIn the formula booklet	\(\frac{d}{dx}\Big(\frac{f}{g}\Big)=\frac{f'g-fg'}{g^2}\)
Inverse	\(\frac{dx}{dy}=\frac1{dy/dx}\)
Integrals	\(\int e^{kx}dx=\tfrac1ke^{kx}+c,\) \(\int\tfrac1x\,dx=\ln\|x\|+c,\) \(\int\cos kx\,dx=\tfrac1k\sin kx+c\)
Iteration; sign change	\(x_{n+1}=g(x_n);\) \(f(a)f(b)<0\Rightarrow\text{root in }(a,b)\)
Functions: \(fg(x)=f(g(x))\); \(f^{-1}\) reflects \(f\) in \(y=x\); modulus graphs

Worked example

Write 3 sin x + 4 cos x in the form R sin(x + α), with R > 0 and 0 < α < π/2.

R = √(3² + 4²) = 5
R cos α = 3 and R sin α = 4, so tan α = 4/3

Answer: 5 sin(x + 0.927) (α in radians, 3 s.f.)

Common mistakes

Functions: the domain of an inverse (again), fg versus gf, range
Modulus equations and inequalities: extra solutions
Cancelling terms rather than factors
Fg(x) applied as f first

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Simplify rational expressions by factorising and cancelling, and multiply, divide, add and subtract algebraic fractions such as 2/(x + 1) − 3/(x² − 1).
Define sec, cosec and cot as reciprocals of cos, sin and tan, and find exact values and simplify expressions involving them.
Write a sin x ± b cos x in the forms R sin(x ± α) and R cos(x ± α), finding R and α exactly or to a given accuracy.
Use logarithms to turn y = axⁿ and y = kbˣ into straight-line graphs, and find the constants from the gradient and intercept.
Use the chain, product and quotient rules to find tangents, normals and stationary points of curves that combine exponential, logarithmic and trigonometric functions.
Draw staircase and cobweb diagrams to show whether an iteration converges, and explain why some rearrangements of an equation fail to converge.

The printable sheet

Pure Mathematics 3 (P3) knowledge organiser for Edexcel IAL Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Pure Mathematics 3 (P3) knowledge organiser (Edexcel IAL Maths), A4. Download the PDF.

Revise it next

Other Edexcel IAL Maths topics: Pure Mathematics 1 (P1) · Pure Mathematics 2 (P2) · Pure Mathematics 4 (P4) · Statistics 1 (S1) · Mechanics 1 (M1) · All Edexcel IAL Maths organisers