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Mathematical Formulae - A-Levels Maths

Students must be able to use the following formulae and identities for AS and A-level mathematics, without these formulae and identities being provided, either in these forms or in equivalent forms. These formulae and identities may only be provided where they are the starting point for a proof or as a result to be proved.

Pure mathematics

Quadratic equations

Roots of a quadratic equation
ax2+bx+c=0⇒x=−b±b2−4ac2a\boldsymbol{ax^2+bx+c=0 \quad\Rightarrow\quad x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}

Laws of indices

Multiplication
axay=ax+y\boldsymbol{a^x a^y = a^{x+y}}
Division
ax÷ay=ax−y\boldsymbol{a^x \div a^y = a^{x-y}}
Power of a power
(ax)y=axy\boldsymbol{(a^x)^y = a^{xy}}

Laws of logarithms

Definition
x=an  ⟺  n=log⁡ax(a>0, x>0)\boldsymbol{x=a^n \iff n=\log_a x \quad (a>0,\ x>0)}
Product law
log⁡ax+log⁡ay=log⁡a(xy)\boldsymbol{\log_a x+\log_a y=\log_a(xy)}
Quotient law
log⁡ax−log⁡ay=log⁡a(xy)\boldsymbol{\log_a x-\log_a y=\log_a\left(\frac{x}{y}\right)}
Power law
klog⁡ax=log⁡a(xk)\boldsymbol{k\log_a x=\log_a(x^k)}

Coordinate geometry

A straight line graph, gradient m passing through (x₁, y₁), has equation:

Equation of a straight line
y−y1=m(x−x1)\boldsymbol{y-y_1=m(x-x_1)}
Perpendicular gradients
m1m2=−1\boldsymbol{m_1m_2=-1}

Sequences

General term of an arithmetic progression
un=a+(n−1)d\boldsymbol{u_n=a+(n-1)d}
General term of a geometric progression
un=arn−1\boldsymbol{u_n=ar^{n-1}}

Trigonometry

In the triangle ABC\boldsymbol{ABC}

Trigonometric formulae and identities

Sine rule
asin⁡A=bsin⁡B=csin⁡C\boldsymbol{\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}}
Cosine rule
a2=b2+c2−2bccos⁡A\boldsymbol{a^2=b^2+c^2-2bc\cos A}
Area of a triangle
Area=12absin⁡C\boldsymbol{\text{Area}=\frac{1}{2}ab\sin C}
Pythagorean identity
cos⁡2A+sin⁡2A=1\boldsymbol{\cos^2 A+\sin^2 A=1}
Secant identity
sec⁡2A=1+tan⁡2A\boldsymbol{\sec^2 A=1+\tan^2 A}
Cosecant identity
cosec⁡2A=1+cot⁡2A\boldsymbol{\operatorname{cosec}^2 A=1+\cot^2 A}
Double-angle sine
sin⁡2A=2sin⁡Acos⁡A\boldsymbol{\sin 2A=2\sin A\cos A}
Double-angle cosine
cos⁡2A=cos⁡2A−sin⁡2A\boldsymbol{\cos 2A=\cos^2 A-\sin^2 A}
Double-angle tangent
tan⁡2A=2tan⁡A1−tan⁡2A\boldsymbol{\tan 2A=\frac{2\tan A}{1-\tan^2 A}}

Mensuration

Circle

Circumference and area of a circle, radius r and diameter d:

Circumference
C=2πr=πd\boldsymbol{C=2\pi r=\pi d}
Area
A=πr2\boldsymbol{A=\pi r^2}

Pythagoras’ theorem

In any right-angled triangle where a, b and c are the lengths of the sides and c is the hypotenuse:

Pythagoras’ theorem
c2=a2+b2\boldsymbol{c^2=a^2+b^2}

Area and volume

Area of a trapezium
A=12(a+b)h\boldsymbol{A=\frac{1}{2}(a+b)h}
Volume of a prism
V=area of cross-section×length\boldsymbol{V=\text{area of cross-section}\times\text{length}}

Arc length and sector area

For a circle of radius r, where an angle at the centre of θ radians subtends an arc of length s and encloses an associated sector of area A:

Arc length
s=rθ\boldsymbol{s=r\theta}
Area of a sector
A=12r2θ\boldsymbol{A=\frac{1}{2}r^2\theta}

Calculus and differential equations

Differentiation

FunctionDerivative
xn\boldsymbol{x^n}nxn−1\boldsymbol{nx^{n-1}}
sin⁡kx\boldsymbol{\sin kx}kcos⁡kx\boldsymbol{k\cos kx}
cos⁡kx\boldsymbol{\cos kx}−ksin⁡kx\boldsymbol{-k\sin kx}
ekx\boldsymbol{e^{kx}}kekx\boldsymbol{ke^{kx}}
ln⁡x\boldsymbol{\ln x}1x\boldsymbol{\frac{1}{x}}
f(x)+g(x)\boldsymbol{f(x)+g(x)}f′(x)+g′(x)\boldsymbol{f'(x)+g'(x)}
f(x)g(x)\boldsymbol{f(x)g(x)}f′(x)g(x)+f(x)g′(x)\boldsymbol{f'(x)g(x)+f(x)g'(x)}
f(g(x))\boldsymbol{f(g(x))}f′(g(x))g′(x)\boldsymbol{f'(g(x))g'(x)}

Integration

FunctionIntegral
xn\boldsymbol{x^n}1n+1xn+1+c,n≠−1\boldsymbol{\frac{1}{n+1}x^{n+1}+c,\quad n\ne-1}
cos⁡kx\boldsymbol{\cos kx}1ksin⁡kx+c\boldsymbol{\frac{1}{k}\sin kx+c}
sin⁡kx\boldsymbol{\sin kx}−1kcos⁡kx+c\boldsymbol{-\frac{1}{k}\cos kx+c}
ekx\boldsymbol{e^{kx}}1kekx+c\boldsymbol{\frac{1}{k}e^{kx}+c}
1x\boldsymbol{\frac{1}{x}}ln⁡∣x∣+c,x≠0\boldsymbol{\ln|x|+c,\quad x\ne0}
f′(x)+g′(x)\boldsymbol{f'(x)+g'(x)}f(x)+g(x)+c\boldsymbol{f(x)+g(x)+c}
f′(g(x))g′(x)\boldsymbol{f'(g(x))g'(x)}f(g(x))+c\boldsymbol{f(g(x))+c}

Area under a curve

Area under a curve
Area under a curve=∫aby dx(y≥0)\boldsymbol{\text{Area under a curve}=\int_a^b y\,dx\quad(y\ge0)}

Vectors

Magnitude of a vector

Vector magnitude
∣xi+yj+zk∣=x2+y2+z2\boldsymbol{|x\mathbf{i}+y\mathbf{j}+z\mathbf{k}|=\sqrt{x^2+y^2+z^2}}

Mechanics

Forces and equilibrium

Weight
Weight=mass×g\boldsymbol{\text{Weight}=\text{mass}\times g}
Friction
F≤μR\boldsymbol{F\le\mu R}
Newton’s second law
F=ma\boldsymbol{F=ma}

Kinematics

For motion in a straight line with variable acceleration:

Velocity
v=drdt\boldsymbol{v=\frac{dr}{dt}}
Acceleration
a=dvdt=d2rdt2\boldsymbol{a=\frac{dv}{dt}=\frac{d^2r}{dt^2}}
Displacement
r=∫v dt\boldsymbol{r=\int v\,dt}
Velocity from acceleration
v=∫a dt\boldsymbol{v=\int a\,dt}

Statistics

Statistical formulae

Mean of a set of data
xˉ=∑xn=∑fx∑f\boldsymbol{\bar{x}=\frac{\sum x}{n}=\frac{\sum fx}{\sum f}}
Standard Normal variable
Z=X−μσwhereX∼N(μ,σ2)\boldsymbol{Z=\frac{X-\mu}{\sigma}\quad\text{where}\quad X\sim N(\mu,\sigma^2)}