Formula sheet

The formulas behind Kalkuli's calculators, grouped by topic. Each group links to the tool or guide that uses it.

By Published Updated

Algebra

Expanding brackets
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²
Quadratic formula
ax² + bx + c = 0
x = (−b ± √(b² − 4ac)) / 2a
Discriminant Δ = b² − 4ac. Solver
Roots of a quadratic
x₁ + x₂ = −b/a
x₁ × x₂ = c/a
Vertex at x = −b/2a

Trigonometry

Right-angled triangles
sin θ = opp / hyp
cos θ = adj / hyp
tan θ = opp / adj
Pythagorean identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
Exact values
sin 30° = ½, cos 30° = √3/2
sin 45° = cos 45° = √2/2
tan 45° = 1, tan 60° = √3
Addition formulas
sin(α ± β) = sin α cos β ± cos α sin β
cos(α ± β) = cos α cos β ∓ sin α sin β
Double angle
sin 2θ = 2 sin θ cos θ
cos 2θ = 1 − 2sin²θ = 2cos²θ − 1
tan 2θ = 2tan θ / (1 − tan²θ)
Any triangle
a/sin A = b/sin B = c/sin C
c² = a² + b² − 2ab cos C
Area = ½ab sin C
Solver · Which rule?
Degrees and radians
rad = deg × π/180
deg = rad × 180/π
Guide

Logarithms and exponentials

Definition
log_b(x) = y ⟺ bʸ = x
ln x = logₑ x
Laws
log(xy) = log x + log y
log(x/y) = log x − log y
log(xᵏ) = k log x
Change of base
log_b(x) = ln x / ln b
e.g. log₂ 10 = ln 10 / ln 2 ≈ 3.3219
Index laws
xᵃ · xᵇ = xᵃ⁺ᵇ
xᵃ / xᵇ = xᵃ⁻ᵇ
(xᵃ)ᵇ = xᵃᵇ, x⁰ = 1

Areas and volumes

Circle
Area = πr²
Circumference = 2πr
Arc length = rθ (θ in radians)
Triangle (Heron)
s = (a + b + c)/2
Area = √(s(s−a)(s−b)(s−c))
Sphere, cylinder, cone
Sphere V = 4/3 πr³, A = 4πr²
Cylinder V = πr²h
Cone V = 1/3 πr²h

Statistics and counting

Mean and standard deviation
x̄ = Σx / n
s = √(Σ(x − x̄)² / (n − 1))
σ = √(Σ(x − μ)² / n)
Worked example
Standard error
SE = s / √n
Permutations and combinations
nPr = n! / (n − r)!
nCr = n! / (r!(n − r)!)
Calculator

Matrices (2×2)

Determinant
det [[a, b], [c, d]] = ad − bc
Inverse
A⁻¹ = 1/(ad − bc) × [[d, −b], [−c, a]]
3×3 method