Mathematics
Algebra, trigonometry, sequences, calculus, vectors and statistics.
60 formulas
Algebra & Functions
Quadratic Formula
Solves the general quadratic ax² + bx + c = 0 for both roots when a ≠ 0.
Discriminant
Sign of Δ classifies the roots of ax² + bx + c = 0: Δ>0 two real, Δ=0 repeated, Δ<0 complex.
Completing the square
Rewrites a quadratic in vertex form (x + p)² + q to read off the turning point (−p, q).
Logarithm product law
Multiplication inside a log becomes addition outside (base a > 0, a ≠ 1, x, y > 0).
Exponential growth/decay
Continuous exponential model: k > 0 grows, k < 0 decays. Used for population, radioactive decay, cooling.
Power rule (derivative)
Derivative of xⁿ for any real n. Foundation of polynomial differentiation.
Power rule (integral)
Indefinite integral of xⁿ. The +C constant is essential.
Binomial coefficient
Binomial coefficient
Sequences & Series
Arithmetic sequence — nth term
n-th term of an arithmetic progression with first term a and common difference d.
Arithmetic series — sum of n terms
Sum of first n terms of an arithmetic progression. Equivalent form: Sₙ = n(a + l)/2 where l is the last term.
Geometric sequence — nth term
n-th term of a geometric progression with first term a and common ratio r.
Geometric series — sum of n terms
Sum of first n terms of a geometric progression.
Geometric series — sum to infinity
A geometric series converges to a/(1 − r) only when |r| < 1.
Geometric series sum
Sum of the first n terms of a geometric sequence (r ≠ 1).
AP nth term
Sine rule relates each side of a triangle to the sine of its opposite angle, used to find unknown sides or angles in non-right triangles.
Arithmetic series sum
Sum of the first n terms of an arithmetic progression with first term a and common difference d.
GP nth term
GP nth term
GP sum (finite)
GP sum (finite)
GP sum to infinity
GP sum to infinity
Trigonometry
Pythagoras' Theorem
In a right-angled triangle, the square of the hypotenuse c equals the sum of the squares of the other two sides.
Sine Rule
Solves non-right triangles when an angle and its opposite side are paired with one further side or angle.
Cosine Rule
Generalises Pythagoras to any triangle — use when two sides + included angle, or all three sides, are known.
Area of triangle (sine)
Area of any triangle using two sides and the included angle.
Arc length (radians)
Arc length of a circle sector with radius r and angle θ in radians.
Sector area (radians)
Area of a circular sector with radius r and angle θ in radians.
Arc length (radians)
Arc length of a circular sector with angle in radians.
Sine rule
Sine rule: each side of any triangle is proportional to the sine of its opposite angle — used to find unknown sides or angles in non-right triangles.
Cosine rule
Cosine rule
Triangle area
Triangle area
Arc length
Arc length
Sector area
Sector area
Volume of cylinder
Volume of cylinder
Volume of sphere
Volume of sphere
Surface area of sphere
Surface area of sphere
Volume of cone
Volume of cone
Pythagoras
Pythagoras
Coordinate Geometry
Distance between two points
Euclidean distance between (x₁,y₁) and (x₂,y₂) in the plane.
Midpoint of a segment
Midpoint of the segment joining (x₁,y₁) and (x₂,y₂).
Gradient of a line
Slope between two points; undefined when x₁ = x₂ (vertical line).
Circle equation
Circle with centre (a, b) and radius r. Distance from any point on the circle to the centre equals r.
Distance between two points
Euclidean distance between two coordinates in 2D.
Midpoint x-coordinate
x-coordinate of the midpoint of a line segment.
Gradient between two points
Slope of the line through two points.
Circle equation
Circle equation
Calculus
Vectors
Statistics & Probability
Percentage change
Relative change between two values, expressed as a percentage. Negative result indicates decrease.
Simple interest
Interest paid only on the original principal P at annual rate r (decimal) over t years.
Compound interest
Final amount A when principal P grows at rate r per period (decimal) for t periods, compounding once per period.
Arithmetic mean
Mean of n values whose total is Σx.
Variance (population)
Population variance via the computational formula. Square root gives the standard deviation σ.
Binomial probability
Probability of exactly k successes in n independent Bernoulli trials with success probability p.
Poisson probability
Probability of k events occurring in a fixed interval when events occur independently at average rate λ.
Conditional probability
Probability of A given B has occurred, when P(B) > 0.
Standardising (z-score)
Number of standard deviations a value x lies from the mean μ. Used to look up normal probabilities.
Combinations nCr
Combinations nCr
Permutations nPr
Permutations nPr
Z-score (standard score)
Number of standard deviations a value x lies from the mean of its distribution.