Higher Maths formula sheet (SQA)

m = (y₂ − y₁) / (x₂ − x₁); m = tan θ

θ is the angle the line makes with the positive direction of the x-axis.

y − b = m(x − a)

Through the point (a, b) with gradient m. Also y = mx + c and ax + by + c = 0.

Parallel: m₁ = m₂; perpendicular: m₁ × m₂ = −1

d = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Use the midpoint and perpendicular gradient for perpendicular bisectors; a median joins a vertex to the midpoint of the opposite side; an altitude is perpendicular to the opposite side.

A tangent is perpendicular to the radius at the point of contact. Substitute the line into the circle: discriminant = 0 ⇒ tangent; > 0 ⇒ two points; < 0 ⇒ no intersection

|a| = √(a₁² + a₂² + a₃²); a unit vector has magnitude 1

a·b = 0 ⇔ a and b are perpendicular (a, b non-zero)

A, B and C are collinear if AB = k·BC with B a common point; if P divides AB in the ratio m : n, then p = (n·a + m·b)/(m + n)

Bold letters are position vectors.

For ax² + bx + c = 0: b² − 4ac > 0 two real roots; = 0 equal roots; < 0 no real roots

If f(a) = 0 then (x − a) is a factor of f(x); the remainder when f(x) is divided by (x − a) is f(a)

Use synthetic division to find f(a) and the quotient together.

log_a xy = log_a x + log_a y; log_a (x/y) = log_a x − log_a y; log_a xⁿ = n log_a x
log_a 1 = 0; log_a a = 1; y = a^x ⇔ x = log_a y

ln x = log_e x. Use logs to find a and b from straight-line graphs of y = ab^x or y = ax^b.

u_n+1 = au_n + b; limit L = b / (1 − a), which exists when −1 < a < 1

sin²x + cos²x = 1; tan x = sin x / cos x; π radians = 180°

sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2, tan 45° = 1; sin 60° = √3/2, cos 60° = ½, tan 60° = √3

Learn them in radians too: π/6, π/4 and π/3.

a cos x + b sin x = k cos(x − α), where k = √(a² + b²)

Expand k cos(x − α) = k cos x cos α + k sin x sin α, compare coefficients, then use the signs of k cos α and k sin α to find the quadrant of α. The same method works for k sin(x ± α) and k cos(x + α).

d/dx (xⁿ) = nxⁿ⁻¹; d/dx [f(g(x))] = f′(g(x)) × g′(x)

Write roots and fractions as powers first, for example 1/√x = x^−½.

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + c (n ≠ −1); ∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / (a(n + 1)) + c

Gradient of the tangent at x = a is f′(a); stationary points where f′(x) = 0, then use a nature table

Area between curves = ∫_a^b (upper − lower) dx

Area below the x-axis gives a negative integral: split the limits and take the size of each part.