O-Level A-Math formula sheet

The A-Math formula list covers the quadratic formula, binomial expansion and trigonometric identities. Everything else, including all of differentiation and integration, you must memorise.

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Given in the exam

Printed at the front of Paper 1 and Paper 2.

Quadratic equationFor the equation

ax^{2} + bx + c = 0

x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

Binomial expansion

(a + b)^{n} = a^{n} + \dbinom{n}{1}a^{n-1}b + \dbinom{n}{2}a^{n-2}b^{2} + \ldots + \dbinom{n}{r}a^{n-r}b^{r} + \ldots + b^{n}

,where

n

is a positive integer and

\dbinom{n}{r} = \dfrac{n!}{r!\,(n-r)!} = \dfrac{n(n-1)\ldots(n-r+1)}{r!}

Identities

\sin^{2} A + \cos^{2} A = 1

\sec^{2} A = 1 + \tan^{2} A

\operatorname{cosec}^{2} A = 1 + \cot^{2} A

\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) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

\sin 2A = 2\sin A\cos A

\cos 2A = \cos^{2} A - \sin^{2} A

\cos 2A = 2\cos^{2} A - 1

\cos 2A = 1 - 2\sin^{2} A

\tan 2A = \dfrac{2\tan A}{1 - \tan^{2} A}

Formulae for

\triangle ABC

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

a^{2} = b^{2} + c^{2} - 2bc\cos A

\Delta = \dfrac{1}{2}bc\sin A

Formulas to memorise

Quadratic functions

NameFormulaExample

Two real, distinct roots

b^2 - 4ac > 0

x^2 - 5x + 6

25 - 24 = 1 > 0

Two real, equal roots

b^2 - 4ac = 0

x^2 + 4x + 4

16 - 16 = 0

No real roots

b^2 - 4ac < 0

x^2 + 2x + 5

4 - 20 < 0

Always positive

a > 0

and

b^2 - 4ac < 0

x^2 + 2x + 5

is always positive

Turning point

y = a(x - h)^2 + k

: turning point

(h, k)

y = 2(x - 1)^2 + 3

: minimum point

(1, 3)

Surds

NameFormulaExample

Multiplying

\sqrt{a} \times \sqrt{b} = \sqrt{ab}

\sqrt{2} \times \sqrt{8} = 4

Rationalising

\dfrac{1}{\sqrt{a} + \sqrt{b}} = \dfrac{\sqrt{a} - \sqrt{b}}{a - b}

\dfrac{1}{\sqrt{3} + \sqrt{2}} = \sqrt{3} - \sqrt{2}

Polynomials and binomial theorem

NameFormulaExample

Remainder theorem

f(x) \div (x - a)

leaves remainder

f(a)

x^3 - 7x + 4

x - 2

: remainder

-2

Factor theorem

f(a) = 0 \iff (x - a)

is a factor

x^3 - 6x^2 + 11x - 6

f(1) = 0

, so

x - 1

is a factor

General term

T_{r+1} = \dbinom{n}{r} a^{n-r} b^{r}

Coefficient of

x^2

(1 + 2x)^5

\dbinom{5}{2} \times 2^2 = 40

Exponentials and logarithms

NameFormulaExample

Definition

a^x = b \iff x = \log_a b

2^x = 16

x = 4

Product law

\log_a xy = \log_a x + \log_a y

\lg 2 + \lg 5 = \lg 10 = 1

Quotient law

\log_a \dfrac{x}{y} = \log_a x - \log_a y

\log_3 54 - \log_3 2 = \log_3 27 = 3

Power law

\log_a x^r = r\log_a x

\log_2 32 = 5\log_2 2 = 5

Change of base

\log_a b = \dfrac{\log_c b}{\log_c a}

\log_4 8 = \dfrac{\log_2 8}{\log_2 4} = \dfrac{3}{2}

Special values

\log_a a = 1

\log_5 5 = 1

Special values

\log_a 1 = 0

\ln 1 = 0

e and ln

\ln e^{x} = x

\ln e^{3} = 3

Coordinate geometry

NameFormulaExample

Midpoint

\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)

(1, 2)

and

(5, 8)

(3, 5)

Perpendicular lines

m_1 m_2 = -1

m_1 = 2

m_2 = -\dfrac{1}{2}

Circle

(x - a)^2 + (y - b)^2 = r^2

(x - 3)^2 + (y + 1)^2 = 25

: centre

(3, -1)

, radius

5

Circle (general form)

x^2 + y^2 + 2gx + 2fy + c = 0

: centre

(-g, -f)

, radius

\sqrt{g^2 + f^2 - c}

x^2 + y^2 - 4x + 6y - 12 = 0

: centre

(2, -3)

, radius

5

Area of a polygon

\dfrac{1}{2}\left|x_1y_2 + x_2y_3 + x_3y_1 - x_2y_1 - x_3y_2 - x_1y_3\right|

(0, 0)

(4, 0)

(0, 3)

6

Trigonometry

NameFormulaExample

Tangent

\tan A = \dfrac{\sin A}{\cos A}

\tan 45^\circ = \dfrac{\sin 45^\circ}{\cos 45^\circ} = 1

Exact value

\sin 30^\circ = \dfrac{1}{2}

\cos 60^\circ = \dfrac{1}{2}

Exact value

\sin 45^\circ = \dfrac{\sqrt{2}}{2}

\cos 45^\circ = \dfrac{\sqrt{2}}{2}

Exact value

\sin 60^\circ = \dfrac{\sqrt{3}}{2}

\tan 60^\circ = \sqrt{3}

R-formula

a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)

R = \sqrt{a^2 + b^2}

\tan\alpha = \dfrac{b}{a}

3\sin\theta + 4\cos\theta = 5\sin(\theta + 53.1^\circ)

Radians

\pi \text{ rad} = 180^\circ

60^\circ = \dfrac{\pi}{3}

rad

Arc length

s = r\theta

(

\theta

in radians)

r = 6

\theta = 1.5

s = 9

Sector area

A = \dfrac{1}{2}r^2\theta

(

\theta

in radians)

r = 6

\theta = 1.5

A = 27

Differentiation

NameFormulaExample

Power rule

\dfrac{d}{dx}x^n = nx^{n-1}

\dfrac{d}{dx}x^5 = 5x^4

Chain rule

\dfrac{d}{dx}(ax + b)^n = an(ax + b)^{n-1}

\dfrac{d}{dx}(3x + 1)^4 = 12(3x + 1)^3

Product rule

\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}

\dfrac{d}{dx}(x^2\sin x) = 2x\sin x + x^2\cos x

Quotient rule

\dfrac{d}{dx}\left(\dfrac{u}{v}\right) = \dfrac{v\dfrac{du}{dx} - u\dfrac{dv}{dx}}{v^2}

\dfrac{d}{dx}\left(\dfrac{x}{x + 1}\right) = \dfrac{1}{(x + 1)^2}

Sine

\dfrac{d}{dx}\sin x = \cos x

\dfrac{d}{dx}\sin 3x = 3\cos 3x

Cosine

\dfrac{d}{dx}\cos x = -\sin x

\dfrac{d}{dx}\cos 2x = -2\sin 2x

Tangent

\dfrac{d}{dx}\tan x = \sec^2 x

\dfrac{d}{dx}\tan 4x = 4\sec^2 4x

Exponential

\dfrac{d}{dx}e^{x} = e^{x}

\dfrac{d}{dx}e^{5x} = 5e^{5x}

Logarithm

\dfrac{d}{dx}\ln x = \dfrac{1}{x}

\dfrac{d}{dx}\ln(2x + 1) = \dfrac{2}{2x + 1}

Stationary point

\dfrac{dy}{dx} = 0

; minimum if

\dfrac{d^2y}{dx^2} > 0

, maximum if

\dfrac{d^2y}{dx^2} < 0

y = x^2 - 4x

x = 2

, minimum

Connected rates

\dfrac{dy}{dt} = \dfrac{dy}{dx} \times \dfrac{dx}{dt}

y = x^2

\dfrac{dx}{dt} = 3

x = 2

\dfrac{dy}{dt} = 12

Integration

NameFormulaExample

Power rule

\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n + 1} + c

n \ne -1

\displaystyle\int x^3\,dx = \dfrac{x^4}{4} + c

Linear bracket

\displaystyle\int (ax + b)^n\,dx = \dfrac{(ax + b)^{n+1}}{a(n + 1)} + c

\displaystyle\int (2x + 1)^3\,dx = \dfrac{(2x + 1)^4}{8} + c

Sine

\displaystyle\int \sin(ax + b)\,dx = -\dfrac{1}{a}\cos(ax + b) + c

\displaystyle\int \sin 2x\,dx = -\dfrac{1}{2}\cos 2x + c

Cosine

\displaystyle\int \cos(ax + b)\,dx = \dfrac{1}{a}\sin(ax + b) + c

\displaystyle\int \cos 3x\,dx = \dfrac{1}{3}\sin 3x + c

Secant squared

\displaystyle\int \sec^2(ax + b)\,dx = \dfrac{1}{a}\tan(ax + b) + c

\displaystyle\int \sec^2 x\,dx = \tan x + c

Exponential

\displaystyle\int e^{ax + b}\,dx = \dfrac{1}{a}e^{ax + b} + c

\displaystyle\int e^{3x + 1}\,dx = \dfrac{1}{3}e^{3x + 1} + c

Reciprocal

\displaystyle\int \dfrac{1}{ax + b}\,dx = \dfrac{1}{a}\ln(ax + b) + c

\displaystyle\int \dfrac{1}{2x + 1}\,dx = \dfrac{1}{2}\ln(2x + 1) + c

Definite integral

\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)

\displaystyle\int_0^2 x^2\,dx = \dfrac{8}{3}

Kinematics

NameFormulaExample

Velocity

v = \dfrac{ds}{dt}

s = t^3 - 3t

v = 3t^2 - 3

Acceleration

a = \dfrac{dv}{dt}

v = 3t^2 - 3

a = 6t

Displacement from velocity

s = \displaystyle\int v\,dt

v = 2t

s = 0

t = 0

s = t^2

Practise using them

Given in the exam

Printed at the front of Paper 1 and Paper 2.

Quadratic equationFor the equation

ax^{2} + bx + c = 0

x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

Binomial expansion

(a + b)^{n} = a^{n} + \dbinom{n}{1}a^{n-1}b + \dbinom{n}{2}a^{n-2}b^{2} + \ldots + \dbinom{n}{r}a^{n-r}b^{r} + \ldots + b^{n}

,where

n

is a positive integer and

\dbinom{n}{r} = \dfrac{n!}{r!\,(n-r)!} = \dfrac{n(n-1)\ldots(n-r+1)}{r!}

Identities

\sin^{2} A + \cos^{2} A = 1

\sec^{2} A = 1 + \tan^{2} A

\operatorname{cosec}^{2} A = 1 + \cot^{2} A

\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) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

\sin 2A = 2\sin A\cos A

\cos 2A = \cos^{2} A - \sin^{2} A

\cos 2A = 2\cos^{2} A - 1

\cos 2A = 1 - 2\sin^{2} A

\tan 2A = \dfrac{2\tan A}{1 - \tan^{2} A}

Formulae for

\triangle ABC

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

a^{2} = b^{2} + c^{2} - 2bc\cos A

\Delta = \dfrac{1}{2}bc\sin A

Formulas to memorise

Quadratic functions

NameFormulaExample

Two real, distinct roots

b^2 - 4ac > 0

x^2 - 5x + 6

25 - 24 = 1 > 0

Two real, equal roots

b^2 - 4ac = 0

x^2 + 4x + 4

16 - 16 = 0

No real roots

b^2 - 4ac < 0

x^2 + 2x + 5

4 - 20 < 0

Always positive

a > 0

and

b^2 - 4ac < 0

x^2 + 2x + 5

is always positive

Turning point

y = a(x - h)^2 + k

: turning point

(h, k)

y = 2(x - 1)^2 + 3

: minimum point

(1, 3)

Surds

NameFormulaExample

Multiplying

\sqrt{a} \times \sqrt{b} = \sqrt{ab}

\sqrt{2} \times \sqrt{8} = 4

Rationalising

\dfrac{1}{\sqrt{a} + \sqrt{b}} = \dfrac{\sqrt{a} - \sqrt{b}}{a - b}

\dfrac{1}{\sqrt{3} + \sqrt{2}} = \sqrt{3} - \sqrt{2}

Polynomials and binomial theorem

NameFormulaExample

Remainder theorem

f(x) \div (x - a)

leaves remainder

f(a)

x^3 - 7x + 4

x - 2

: remainder

-2

Factor theorem

f(a) = 0 \iff (x - a)

is a factor

x^3 - 6x^2 + 11x - 6

f(1) = 0

, so

x - 1

is a factor

General term

T_{r+1} = \dbinom{n}{r} a^{n-r} b^{r}

Coefficient of

x^2

(1 + 2x)^5

\dbinom{5}{2} \times 2^2 = 40

Exponentials and logarithms

NameFormulaExample

Definition

a^x = b \iff x = \log_a b

2^x = 16

x = 4

Product law

\log_a xy = \log_a x + \log_a y

\lg 2 + \lg 5 = \lg 10 = 1

Quotient law

\log_a \dfrac{x}{y} = \log_a x - \log_a y

\log_3 54 - \log_3 2 = \log_3 27 = 3

Power law

\log_a x^r = r\log_a x

\log_2 32 = 5\log_2 2 = 5

Change of base

\log_a b = \dfrac{\log_c b}{\log_c a}

\log_4 8 = \dfrac{\log_2 8}{\log_2 4} = \dfrac{3}{2}

Special values

\log_a a = 1

\log_5 5 = 1

Special values

\log_a 1 = 0

\ln 1 = 0

e and ln

\ln e^{x} = x

\ln e^{3} = 3

Coordinate geometry

NameFormulaExample

Midpoint

\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)

(1, 2)

and

(5, 8)

(3, 5)

Perpendicular lines

m_1 m_2 = -1

m_1 = 2

m_2 = -\dfrac{1}{2}

Circle

(x - a)^2 + (y - b)^2 = r^2

(x - 3)^2 + (y + 1)^2 = 25

: centre

(3, -1)

, radius

5

Circle (general form)

x^2 + y^2 + 2gx + 2fy + c = 0

: centre

(-g, -f)

, radius

\sqrt{g^2 + f^2 - c}

x^2 + y^2 - 4x + 6y - 12 = 0

: centre

(2, -3)

, radius

5

Area of a polygon

\dfrac{1}{2}\left|x_1y_2 + x_2y_3 + x_3y_1 - x_2y_1 - x_3y_2 - x_1y_3\right|

(0, 0)

(4, 0)

(0, 3)

6

Trigonometry

NameFormulaExample

Tangent

\tan A = \dfrac{\sin A}{\cos A}

\tan 45^\circ = \dfrac{\sin 45^\circ}{\cos 45^\circ} = 1

Exact value

\sin 30^\circ = \dfrac{1}{2}

\cos 60^\circ = \dfrac{1}{2}

Exact value

\sin 45^\circ = \dfrac{\sqrt{2}}{2}

\cos 45^\circ = \dfrac{\sqrt{2}}{2}

Exact value

\sin 60^\circ = \dfrac{\sqrt{3}}{2}

\tan 60^\circ = \sqrt{3}

R-formula

a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)

R = \sqrt{a^2 + b^2}

\tan\alpha = \dfrac{b}{a}

3\sin\theta + 4\cos\theta = 5\sin(\theta + 53.1^\circ)

Radians

\pi \text{ rad} = 180^\circ

60^\circ = \dfrac{\pi}{3}

rad

Arc length

s = r\theta

(

\theta

in radians)

r = 6

\theta = 1.5

s = 9

Sector area

A = \dfrac{1}{2}r^2\theta

(

\theta

in radians)

r = 6

\theta = 1.5

A = 27

Differentiation

NameFormulaExample

Power rule

\dfrac{d}{dx}x^n = nx^{n-1}

\dfrac{d}{dx}x^5 = 5x^4

Chain rule

\dfrac{d}{dx}(ax + b)^n = an(ax + b)^{n-1}

\dfrac{d}{dx}(3x + 1)^4 = 12(3x + 1)^3

Product rule

\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}

\dfrac{d}{dx}(x^2\sin x) = 2x\sin x + x^2\cos x

Quotient rule

\dfrac{d}{dx}\left(\dfrac{u}{v}\right) = \dfrac{v\dfrac{du}{dx} - u\dfrac{dv}{dx}}{v^2}

\dfrac{d}{dx}\left(\dfrac{x}{x + 1}\right) = \dfrac{1}{(x + 1)^2}

Sine

\dfrac{d}{dx}\sin x = \cos x

\dfrac{d}{dx}\sin 3x = 3\cos 3x

Cosine

\dfrac{d}{dx}\cos x = -\sin x

\dfrac{d}{dx}\cos 2x = -2\sin 2x

Tangent

\dfrac{d}{dx}\tan x = \sec^2 x

\dfrac{d}{dx}\tan 4x = 4\sec^2 4x

Exponential

\dfrac{d}{dx}e^{x} = e^{x}

\dfrac{d}{dx}e^{5x} = 5e^{5x}

Logarithm

\dfrac{d}{dx}\ln x = \dfrac{1}{x}

\dfrac{d}{dx}\ln(2x + 1) = \dfrac{2}{2x + 1}

Stationary point

\dfrac{dy}{dx} = 0

; minimum if

\dfrac{d^2y}{dx^2} > 0

, maximum if

\dfrac{d^2y}{dx^2} < 0

y = x^2 - 4x

x = 2

, minimum

Connected rates

\dfrac{dy}{dt} = \dfrac{dy}{dx} \times \dfrac{dx}{dt}

y = x^2

\dfrac{dx}{dt} = 3

x = 2

\dfrac{dy}{dt} = 12

Integration

NameFormulaExample

Power rule

\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n + 1} + c

n \ne -1

\displaystyle\int x^3\,dx = \dfrac{x^4}{4} + c

Linear bracket

\displaystyle\int (ax + b)^n\,dx = \dfrac{(ax + b)^{n+1}}{a(n + 1)} + c

\displaystyle\int (2x + 1)^3\,dx = \dfrac{(2x + 1)^4}{8} + c

Sine

\displaystyle\int \sin(ax + b)\,dx = -\dfrac{1}{a}\cos(ax + b) + c

\displaystyle\int \sin 2x\,dx = -\dfrac{1}{2}\cos 2x + c

Cosine

\displaystyle\int \cos(ax + b)\,dx = \dfrac{1}{a}\sin(ax + b) + c

\displaystyle\int \cos 3x\,dx = \dfrac{1}{3}\sin 3x + c

Secant squared

\displaystyle\int \sec^2(ax + b)\,dx = \dfrac{1}{a}\tan(ax + b) + c

\displaystyle\int \sec^2 x\,dx = \tan x + c

Exponential

\displaystyle\int e^{ax + b}\,dx = \dfrac{1}{a}e^{ax + b} + c

\displaystyle\int e^{3x + 1}\,dx = \dfrac{1}{3}e^{3x + 1} + c

Reciprocal

\displaystyle\int \dfrac{1}{ax + b}\,dx = \dfrac{1}{a}\ln(ax + b) + c

\displaystyle\int \dfrac{1}{2x + 1}\,dx = \dfrac{1}{2}\ln(2x + 1) + c

Definite integral

\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)

\displaystyle\int_0^2 x^2\,dx = \dfrac{8}{3}

Kinematics

NameFormulaExample

Velocity

v = \dfrac{ds}{dt}

s = t^3 - 3t

v = 3t^2 - 3

Acceleration

a = \dfrac{dv}{dt}

v = 3t^2 - 3

a = 6t

Displacement from velocity

s = \displaystyle\int v\,dt

v = 2t

s = 0

t = 0

s = t^2