IGCSE Math Revisionigcsemathrevision.com
Cambridge IGCSE Mathematics (0580)Core & Extended · one-page formula sheet · ★ = not on the paper's formula list
★ Learn it: not on the 0580 list of formulas. On the list printed on the paper. Ext Extended tier only (Papers 2 and 4).
Number
★Indices: \(a^m\times a^n=a^{m+n},\) \(a^m\div a^n=a^{m-n},\) \((a^m)^n=a^{mn},\) \(a^0=1,\) \(a^{-n}=\tfrac1{a^n}\)Use: Simplify and evaluate powers without a calculator
★Fractional indices (Extended only): \(a^{\frac1n}=\sqrt[n]a,\) \(a^{\frac mn}=\big(\sqrt[n]a\big)^m\)Use: Evaluate zero, negative and fractional indices
★Standard form: \(A\times10^n,\) \(1\le A<10,\ n\in\mathbb Z\)Use: Calculate in standard form with and without a calculatorWatch for: The number in front must be at least 1 and less than 10
★Sets: \(n(A\cup B)=n(A)+n(B)\) \({}-n(A\cap B)\)Use: Venn diagrams with two and three setsWatch for: Fill the Venn diagram from the intersection outwards
★Percentage change: \(\frac{\text{new}-\text{original}}{\text{original}}\times100\%\)Use: Percentage increases, decreases and changes
★Reverse percentage (Extended only): \(\text{original}=\frac{\text{new}}{1\pm\frac r{100}}\)Use: Finding the price before a rise or a discount
★Simple interest (rate \(r\)% a year, \(n\) years): \(I=\frac{Prn}{100}\)Use: Simple interest; find the rate or the number of years
★Compound interest: \(A=P\left(1+\tfrac r{100}\right)^n\)Use: Savings, depreciation and population questions
★Growth and decay (Extended only): \(y=a\times k^n:\) \(k>1\text{ growth},\) \(0<k<1\text{ decay}\)Use: Exponential growth and decay, such as depreciation
★Surds (Extended only): \(\sqrt{ab}=\sqrt a\sqrt b,\) \(\frac a{\sqrt b}=\frac{a\sqrt b}b,\) \(\frac1{c+\sqrt b}=\frac{c-\sqrt b}{c^2-b}\)Use: Simplify surds; rationalise denominatorsWatch for: Simplify each surd first, then collect like surds
★Recurring decimal: \(x=0.\dot4\dot5\), so \(100x-x=45\) and \(x=\tfrac{45}{99}=\tfrac5{11}\) (Extended only)Use: Convert recurring decimals into fractions using algebra
★Bounds: a value to the nearest \(u\) lies in \([x-\tfrac u2,\ x+\tfrac u2)\)Use: Rounding and accuracy questions
★Bounds in calculations: for the largest \(a-b\) or \(a\div b\) use the upper bound of \(a\) with the lower bound of \(b\) (Extended only)Use: Greatest possible area or least possible speed
★Rates: \(\text{speed}=\frac{\text{distance}}{\text{time}},\) \(\text{density}=\frac{\text{mass}}{\text{volume}},\) \(\text{pressure}=\frac{\text{force}}{\text{area}}\)Use: Average speed for journeys with several stages
Watch for: Keep every value in the calculator; round only the final answer
Algebra & graphs
Quadratic formula, \(ax^2+bx+c=0\) (Extended only): \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)Use: To a set accuracy or in surd form
★Completing the square (Extended only): \(x^2+bx+c=\left(x+\tfrac b2\right)^2\) \({}-\tfrac{b^2}4+c\)Use: Completed-square form; turning points
★Difference of two squares (Extended only): \(a^2-b^2=(a+b)(a-b)\)Use: Factorise quadratics such as \(9x^{2} - 16\)
★Linear sequence, \(n\)th term: \(u_n=a+(n-1)d\)Use: Finding a rule and a term of a sequence
★Exponential sequence (Extended only): \(u_n=ar^{n-1}\)Use: nth term of exponential sequences such as 3, 6, 12, 24
★Proportion (Extended only): \(y\propto x^n\Rightarrow y\) \({}=kx^n;\) \(y\propto\tfrac1{x^n}\Rightarrow y\) \({}=\tfrac k{x^n}\)Use: Direct and inverse proportion: find the constant
★Functions (Extended only): \(fg(x)=f\big(g(x)\big);\) \(f^{-1}\text{: swap }x,y\text{ and rearrange}\)Use: Composite and inverse functions; solve \(fg(x) = 7\)Watch for: \(gg(x)\) is not \(g(x) \times g(x)\): substitute the whole inner function
★Inequalities: multiplying or dividing by a negative number reverses the signUse: Solve linear inequalities; list integer values
★Straight line: \(y=mx+c\), gradient \(m\), \(y\)-intercept \(c\); vertical line \(x=k\)Use: Equations of straight lines from graphs and descriptions
★Gradient from two points (Extended only): \(m=\frac{y_2-y_1}{x_2-x_1}\)Use: Gradient from a graph or two points
★Parallel lines: \(m_1=m_2\)Use: Line parallel to a given line through a given point
★Perpendicular lines (Extended only): \(m_1\times m_2=-1\)Use: Perpendicular lines and perpendicular bisectors
★Differentiation (Extended only): \(y=ax^n\Rightarrow\frac{dy}{dx}\) \({}=anx^{n-1}\)Use: Gradients of curves and finding maximum and minimum points
★Turning points: \(\frac{dy}{dx}=0\); the gradient of the curve at \(x=a\) is the value of \(\frac{dy}{dx}\) at \(a\) (Extended only)Use: Find stationary points; maximum or minimum
Watch for: Multiply every term in the bracket, including the sign
Geometry & mensuration
★Angles: on a straight line \(180^\circ\); around a point \(360^\circ\); in a triangle \(180^\circ\)Use: Missing angles with a geometric reason for each step
★Polygons (\(n\) sides): \(\text{interior sum}=(n-2)\times180^\circ,\) \(\text{exterior angle (regular)}\) \({}=\tfrac{360^\circ}n\)Use: Interior and exterior angles of polygons
★Circle theorems: angle in a semicircle = 90°; tangent ⟂ radiusUse: Angle in a semicircle and tangent–radius angles
★More circle theorems: angle at centre = 2 × angle at circumference; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; alternate segment theorem; tangents from a point are equal; the perpendicular bisector of a chord passes through the centre (Extended only)Use: Angles at the centre, same segment, cyclic quadrilaterals
Triangle: \(A=\tfrac12bh\)Use: Perimeter and area of triangles
★Parallelogram; trapezium: \(A=bh;\) \(A=\tfrac12(a+b)h\)Use: Area and composite-shape questionsWatch for: The trapezium area is not on the formula list: learn it
Circle: \(A=\pi r^2,\) \(C=2\pi r\)Use: Circumference and area, to a set accuracy or in terms of \(\pi\)
★Arc; sector (angle \(\theta\)): \(\text{arc}=\frac{\theta}{360}\times2\pi r,\) \(\text{sector}=\frac{\theta}{360}\times\pi r^2\)Use: Sectors with the angle in degrees
Curved surface: cylinder; cone (slant \(l\)): \(A=2\pi rh;\) \(A=\pi rl\)Use: Surface areas of cylinders and cones
Sphere, surface area: \(A=4\pi r^2\)Use: Surface area of spheres and hemispheres
Prism; pyramid: \(V=Al;\) \(V=\tfrac13Ah\)Use: Volume of prisms and pyramids
★Similar shapes, scale factor \(k\): lengths \(\times k\)Use: Missing lengths in similar shapes
★Similar shapes: areas \(\times k^2\), volumes \(\times k^3\) (Extended only)Use: Scaling areas and volumes of similar shapes and solids
Watch for: Convert to one unit first: 1 litre \(= 1000 \,\mathrm{cm^{3}}\)
Trigonometry & vectors
★SOHCAHTOA: \(\sin\theta=\frac{\text{opp}}{\text{hyp}},\) \(\cos\theta=\frac{\text{adj}}{\text{hyp}},\) \(\tan\theta=\frac{\text{opp}}{\text{adj}}\)Use: Sides and angles in right-angled triangles, bearings, and elevation
★Bearings: measured clockwise from north, written with 3 figuresUse: Bearings with Pythagoras and trigonometryWatch for: Measure clockwise from north; three figures
★Exact values (Extended only): \(\sin30^\circ=\tfrac12,\) \(\cos60^\circ=\tfrac12,\) \(\tan45^\circ=1,\) \(\sin60^\circ=\tfrac{\sqrt3}2,\) \(\sin45^\circ=\tfrac{\sqrt2}2\)Use: Problems without a calculator using exact values
Sine rule (Extended only): \(\frac a{\sin A}=\frac b{\sin B}\) \({}=\frac c{\sin C}\)Use: Missing sides and angles in any triangle
Cosine rule (Extended only): \(a^2=b^2+c^2-2bc\cos A\)Use: A missing side, or an angle from three sides
Area of a triangle (Extended only): \(\tfrac12ab\sin C\)Use: Area of any triangle; a missing side or angle
★Obtuse angles (Extended only): \(\sin(180^\circ-x)=\sin x,\) \(\cos(180^\circ-x)=-\cos x\)Use: The sine rule when the angle may be obtuse
★Vectors: add and subtract column vectors component by component; \(k\binom xy=\binom{kx}{ky}\) (Extended only)Use: Add, subtract and multiply vectors by scalars
★Magnitude; position vectors (Extended only): \(\left|\binom xy\right|=\sqrt{x^2+y^2},\) \(\overrightarrow{AB}=\mathbf b-\mathbf a\)Use: Vector questions on Papers 2 and 4
Watch for: Non-exact answers to 3 s.f., angles to 1 d.p.
Probability & statistics
★Probability: \(P(\text{not }A)=1-P(A);\) \(\text{expected frequency}\) \({}=n\times P(A)\)Use: Single events and \(P(\text{not}\ A) = 1 - P(A)\)
★Relative frequency: \(\frac{\text{frequency of the outcome}}{\text{number of trials}}\)Use: Estimate probabilities; expected frequencies
★Mutually exclusive: \(P(A\text{ or }B)=P(A)+P(B)\)Use: Combined events from sample spaces and Venn diagramsWatch for: Add only for outcomes that cannot both happen
★Independent: \(P(A\text{ and }B)=P(A)\times P(B)\)Use: Tree diagrams for combined eventsWatch for: For \(A\) and then \(B\) (independent), multiply
★Conditional (Extended only): \(P(A\mid B)=\frac{P(A\cap B)}{P(B)}\)Use: Probability of \(A\) given that \(B\) has happened
★Tree diagrams: multiply along branches, add the outcomes you wantUse: Tree diagrams with and without replacement
★Without replacement: the second-branch probabilities change (one fewer item) (Extended only)Use: Selections without replacement
★Mean from a frequency table: \(\bar x=\frac{\sum fx}{\sum f}\)Use: Mean, median and mode from frequency tables
★Estimated mean, grouped data (class midpoints \(m\)) (Extended only): \(\bar x\approx\frac{\sum fm}{\sum f}\)Use: Estimate the mean of grouped data with midpoints
★Median position (\(n\) values in order): \(\tfrac{n+1}2\text{th value}\)Use: Mean, median, mode and range of discrete data
★Range: \(\text{largest}-\text{smallest}\)Use: Compare two data sets with an average and a spread
★Interquartile range (Extended only): \(Q_3-Q_1\)Use: Compare the spread of two data sets
★Histogram (Extended only): \(\text{frequency density}=\frac{\text{frequency}}{\text{class width}}\)Use: Drawing and reading histograms
Watch for: Along a route multiply; between routes add
