Exact trig values, mensuration, calculus standards and the identities that come up again and again — the formulae worth having instant. Flip on Test yourself to drill recall.
Your board gives some of these and expects the rest from memory — know them all.
GCSE
Exact trig values
sin 0°, 30°, 45°, 60°, 90°
0, ½, √2⁄2, √3⁄2, 1
cos 0°, 30°, 45°, 60°, 90°
1, √3⁄2, √2⁄2, ½, 0
tan 0°, 30°, 45°, 60°
0, 1⁄√3, 1, √3 (tan 90° undefined)
Shape & measure
Circumference of a circle
C = πd = 2πr
Area of a circle
A = πr²
Area of a trapezium
A = ½(a + b)h
Pythagoras' theorem
a² + b² = c²
Trig ratios (SOHCAHTOA)
sin = O⁄H, cos = A⁄H, tan = O⁄A
Volume of a prism
V = area of cross-section × length
Number & algebra
Speed
speed = distance ÷ time
Density
density = mass ÷ volume
Pressure
pressure = force ÷ area
Quadratic formula
x = (−b ± √(b² − 4ac)) ⁄ 2a
A-level
Trig identities
Pythagorean identity
sin²θ + cos²θ ≡ 1
tan in terms of sin, cos
tan θ ≡ sin θ ⁄ cos θ
Secant identity
sec²θ ≡ 1 + tan²θ
Cosecant identity
cosec²θ ≡ 1 + cot²θ
Differentiation (standard results)
Powers
d⁄dx (xⁿ) = n xⁿ⁻¹
sin x
d⁄dx (sin x) = cos x
cos x
d⁄dx (cos x) = −sin x
Exponential
d⁄dx (eˣ) = eˣ
Natural log
d⁄dx (ln x) = 1 ⁄ x
Integration (standard results)
Powers (n ≠ −1)
∫ xⁿ dx = xⁿ⁺¹ ⁄ (n + 1) + c
Reciprocal
∫ 1⁄x dx = ln|x| + c
Exponential
∫ eˣ dx = eˣ + c
cos x
∫ cos x dx = sin x + c
sin x
∫ sin x dx = −cos x + c
Logs, indices & more
Log of a product
logₐ(xy) = logₐx + logₐy
Log of a power
logₐ(xⁿ) = n logₐx
Negative index
a⁻ⁿ = 1 ⁄ aⁿ
Fractional index
a^(1/n) = ⁿ√a
Equation of a circle
(x − a)² + (y − b)² = r²
Discriminant
b² − 4ac (>0 two roots, =0 one, <0 none)
Sharpen the arithmetic underneath
Formulae are only as fast as the mental maths you plug into them. Beat the clock on the trainer.