KS Learning
A Level Maths

Mathematical Equations

graph	equation	conditions
straight line	\(y = x \)	none
quadratic	\(y = x^2 \)	none
cubic	\(y = x^3 \)	none
quartic	\(y = x^4 \)	none
straight line	\(y = - x \)	none
quadratic	\(y = - x^2 \)	none
cubic	\(y = - x^3 \)	none
quartic	\(y = - x^4 \)	none
half quadratic	\( y = { \sqrt{x}} \)	none
quadratic	\( y = {\pm \sqrt{x}} \)	none
hyperbola	\( y = \frac{1}{x} \)	asymptotes \( x = 0 \) and \( y = 0 \)
hyperbola	\(xy = a \)	asymptotes \( x = 0 \) and \( y = 0 \)
hyperbola	\( y = - \frac{1}{x} \)	asymptotes \( x = 0 \) and \( y = 0 \)
hyperbola	\(xy = - a \)	asymptotes \( x = 0 \) and \( y = 0 \)
quadratic	\(y = ax^2 + bx + c \)	none
quadratic	\(y = - ax^2 + bx + c \)	none
quadratic	\(y = (x - a)(x - b) \)	none
cubic	\(y = ax^3 + bx^2 + cx + d \)	none
cubic	\(y = (x - a)(x - b)(x - c) \)	none
circle	\( x^2 + y^2 = r^2 \)	none
circle	\( (x - a)^2 + (y - b)^2 = r^2 \)	none
ellipse	\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)	none
exponential	\(y = a^x \)	asymptote \( y = 0 \)
logarithmic	\(y = log(x) \)	asymptote \( x = 0 \)

Completing the square

Consider the following equation
\(y = 3x^2 - 6x + 5 \)

Factorise the coefficient of \(x^2 \) from the \(x^2 \) term and \(x \) term, in this case 6. Use square brackets to avoid confusion later.
\(y = 3[x^2 - 2x] + 5 \)

Divide the coefficient of \(x \) by 2, in this case -2, and use it to write it inside round brackets that are squared with \(x \).
\(y = 3[(x - 1)^2 \) ...

Subtract the number inside the brackets, square the number, and close the square brackets.
\(y = 3[(x - 1)^2 -(-1)^2] \) ...

Carry down the number on the end of the orignal equation, in this case +5.
\(y = 3[(x - 1)^2 -(-1)^2] + 5 \)

Square the number inside the square brackets, in this case -1.
\(y = 3[(x - 1)^2 -1] + 5 \)

Multiply through by the number outside the square brackets.
\(y = 3(x - 1)^2 -3 + 5 \)

And add the numbers at the end of the equation.
\(y = 3(x - 1)^2 + 2 \)

The job is done!

Trigonometry

1. Definitions

\( sin(x) = \frac{y}{r} \)

\( cos(x) = \frac{x}{r} \)

\( tan(x) = \frac{y}{x} \)

\( sec(x) = \frac{r}{x} \)

\( cosec(x) = \frac{r}{y} \)

\( cot(x) = \frac{x}{y} \)

2. Reciprocals

\( sec(x) = \frac{1}{cos(x)} \)

\( cosec(x) = \frac{1}{sin(x)} \)

\( cot(x) = \frac{1}{tan(x)} \)

3. Identities

\( tan(x) = \frac{sin(x)}{cos(x)} \)

\( cot(x) = \frac{cos(x)}{sin(x)} \)

\( cos^2(x) + sin^2(x) = 1 \)

\( 1 + tan^2(x) = sec^2(x) \)

\( 1 + cot^2(x) = cosec^2(x) \)

4. Triangles

5. Differentials

\( \frac{d}{dx} (sin(x)) = cos(x) \)
\( \frac{d}{dx} (cos(x)) = -sin(x) \)
\( \frac{d}{dx} (tan(x)) = sec^2(x) \)
\( \frac{d}{dx} (sec(x)) = sec(x)tan(x)\)
\( \frac{d}{dx} (cosec(x)) = -cosec(x)cot(x) \)
\( \frac{d}{dx} (cot(x)) = -cosec^2(x) \)

6. Compound Formulae

\( sin(A + B) = sin(A)cos(B) + sin(B)cos(A) \)
\( sin(A - B) = sin(A)cos(B) - sin(B)cos(A) \)
\( cos(A + B) = cos(A)cos(B) - sin(A)sin(B) \)
\( cos(A - B) = cos(A)cos(B) + sin(A)sin(B) \)
\( tan(A + B) = \frac{tan(A) + tan(B)}{1 - tan(A)tan(B)} \)
\( tan(A - B) = \frac{tan(A) - tan(B)}{1 + tan(A)tan(B)} \)

7. Double Angle Formulae

\( sin(2A) = 2sin(A)cos(B) \)
\( cos(2A) = cos(A)^2 - sin(A)^2 \)
\( cos(2A) = 2cos(A)^2 - 1 \)
\( cos(2A) = 1 - 2sin(A)^2 \)
\( tan(2A) = \frac{2tan(A)}{1 - tan(A)^2} \)

8. Graphs

9. Inverse Functions

10. Inverse Differentials

11. Integrals

Mathematical Symbols

Maths makes use of the following symbols and terms

Symbol	Meaning
=	equal to
≡	equivalent to
≠	not equal to
<	less than
>	greater than
≤	less than or equal to
≥	greater than or equal to
∼	tilde operator
≈	approximately equal to
∑	the sum of (greek capital letter sigma)
α	greek small letter alpha
β	greek small letter beta
γ	greek small letter gamma
δ	greek small letter delta
ε	greek small letter epsilon
ζ	greek small letter zeta
η	greek small letter eta
θ	greek small letter theta
λ	greek small letter lambda
μ	greek small letter mu
ν	greek small letter nu
π	reek small letter pi
ρ	greek small letter rho
σ	greek small letter sigma
τ	greek small letter tau
φ	greek small letter phi
χ	greek small letter chi
ψ	greek small letter psi
ω	greek small letter omega
Φ	greek capital letter phi
Χ	greek capital letter chi
Ψ	greek capital letter psi
Ω	greek capital letter omega
∈	element of
∉	not an element of
∏	product sign
∫	integral
∞	infinity
∴	therefore
⇒	therefore
Sum	the result of adding
Difference	the result of substracting
Product	the result of multiplying

SI Prefixes

Prefixes are added to the base unit so 9Gm (giga metres) means 9 x 10⁹ m = 9 000 000 000 m

Index	Symbol	Name
1012	T	tera
109	G	giga
106	M	mega
103	k	kilo
102	h	hecto
101	da	deca
	base unit
10-1	d	deci
10-2	c	centi
10-3	m	milli
10-6	µ	micro
10-9	n	nano
10-12	p	pico