Algebra 5

Exponential calculator y = a · b^x

Heron's formula via the Pythagorean theorem d = (a²−b²+c²)/(2c), h = √(a²−d²), Area = ½·c·h = √(s(s−a)(s−b)(s−c))

Linear equation solver ax + b = 0 → x = −b / a

Logarithm calculator log_b(x) = ln(x) / ln(b)

Quadratic equation solver x = (−b ± √(b²−4ac)) / 2a

Cube Identities 6

Cube of (a−b−c) (a-b-c)³

Cube of a Difference (a−b)³ (a-b)³ = a³ - 3a² b + 3ab² - b³

Cube of a Sum (a+b)³ (a+b)³ = a³ + 3a² b + 3ab² + b³

Cube of a Trinomial (a+b+c)³ (a+b+c)³ = a³+b³+c³+3(a+b)(b+c)(c+a)

Difference (a+b)³ − (a−b)³ (a+b)³ - (a-b)³ = 6a² b + 2b³

Sum (a+b)³ + (a−b)³ (a+b)³ + (a-b)³ = 2a³ + 6ab²

Cube of Linear Binomial 1

Cube of a Linear Binomial (ax+b)³ (ax+b)³ = a³x³ + 3a²bx² + 3ab²x + b³

Difference of Two Squares 2

Difference of Two Squares a²−b² a² - b² = (a-b)(a+b)

General Difference of Squares A²−B² A² - B² = (A-B)(A+B)

Other Useful Identities 6

(a+b)² + (a−b)² = 2(a²+b²) (a+b)² + (a-b)² = 2(a² + b²)

(a+b)² − (a−b)² = 4ab (a+b)² - (a-b)² = 4ab

(ax+b)(cx+d) (ax+b)(cx+d) = acx² + (ad+bc)x + bd

(x+a)(x+b) (x+a)(x+b) = x² + (a+b)x + ab

(x+a)(x−a) (x+a)(x-a) = x² - a²

(x−a)(x−b) (x-a)(x-b) = x² - (a+b)x + ab

Product of Three Terms 1

a³+b³+c³−3abc a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca)

Square Identities 6

Square of (a+b−c) (a+b-c)² = a²+b²+c²+2ab-2bc-2ca

Square of (a−b+c) (a-b+c)² = a²+b²+c²-2ab-2bc+2ca

Square of (a−b−c) (a-b-c)² = a²+b²+c²-2ab-2ca+2bc

Square of a Difference (a−b)² (a-b)² = a² - 2ab + b²

Square of a Sum (a+b)² (a+b)² = a² + 2ab + b²

Square of a Trinomial (a+b+c)² (a+b+c)² = a²+b²+c²+2ab+2bc+2ca

Square of Linear Trinomial 1

Square of a Linear Trinomial (ax+by+c)² (ax+by+c)² = a²x² + b²y² + c² + 2abxy + 2bcy + 2cax

Substitution Identities 4

If x + 1/x = a, then x² + 1/x² = a² − 2 x + 1/x = a => x² + 1/x² = a² - 2

If x + 1/x = a, then x³ + 1/x³ = a³ − 3a x + 1/x = a => x³ + 1/x³ = a³ - 3a

If x − 1/x = a, then x² + 1/x² = a² + 2 x - 1/x = a => x² + 1/x² = a² + 2

If x − 1/x = a, then x³ − 1/x³ = a³ + 3a x - 1/x = a => x³ - 1/x³ = a³ + 3a

Sum / Difference of Two Cubes 2

Difference of Two Cubes a³−b³ a³ - b³ = (a-b)(a² + ab + b²)

Sum of Two Cubes a³+b³ a³ + b³ = (a+b)(a² - ab + b²)

Arithmetic 5

Fraction calculator a/b op c/d → simplest form

GCD (Greatest Common Divisor) calculator gcd via Euclid's algorithm

LCM (Least Common Multiple) calculator lcm(a,b) = |a·b| / gcd(a,b)

Percentage calculator part = (p/100)·whole ; p = (part/whole)·100

Ratio calculator a:b ÷ gcd(a,b) = simplest ratio

2D Shapes 9

Circle Area Calculator area = πr² ; circumference = 2πr ; diameter = 2r

Ellipse Area Calculator area = πab ; circumference ≈ π[3(a+b) − √((3a+b)(a+3b))] (Ramanujan)

Parallelogram Area Calculator area = base×height ; perimeter = 2(base+side)

Rectangle Area Calculator area = length×width ; perimeter = 2(length+width) ; diagonal = √(l²+w²)

Regular Polygon Calculator area = ¼·n·s²·cot(π/n) ; interior angle = (n−2)×180/n

Rhombus Area Calculator area = ½×d₁×d₂ ; side = ½√(d₁²+d₂²) ; perimeter = 4×side

Square Area Calculator area = side² ; perimeter = 4×side ; diagonal = side×√2

Trapezoid Area Calculator area = ½(a+b)×h ; perimeter = a+b+c+d

Triangle Area Calculator ½×base×height, or Heron: √(s(s−a)(s−b)(s−c)) with s = (a+b+c)/2

3D Solids 7

Cone Volume Calculator volume = ⅓πr²h ; slant = √(r²+h²) ; surface = πr(r+slant)

Cube Volume Calculator volume = s³ ; surface area = 6s² ; space diagonal = s√3

Cuboid Volume Calculator volume = l×w×h ; surface area = 2(lw+lh+wh) ; diagonal = √(l²+w²+h²)

Cylinder Volume Calculator volume = πr²h ; lateral = 2πrh ; total = 2πr(r+h)

Sphere Volume Calculator volume = 4/3·πr³ ; surface area = 4πr²

Square Pyramid Volume Calculator volume = ⅓a²h ; slant = √(h²+(a/2)²) ; surface = a² + 2a·slant

Torus Volume Calculator volume = 2π² R r² ; surface area = 4π² R r

Angles 1

Angles & angle types acute < 90° · right = 90° · obtuse 90–180° · straight = 180° · reflex 180–360°

Triangles 2

Heron's formula Area = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2

Pythagoras' theorem a² + b² = c²

Acoustics & Music 1

Sound Beat Frequency f_beat = |f₁ − f₂|; sin(2πf₁t) + sin(2πf₂t)

Astronomy 1

Distance to a star by parallax distance (pc) = 1 / parallax (arcsec)

Construction 1

Roof pitch and rafter length pitch = tan⁻¹(rise / run), rafter = √(rise² + run²)

Electronics 1

AC Circuit Voltage v(t) = V·sin(2πft + φ)

Engineering & Rides 1

Ferris Wheel Height h(t) = R·sin(2πt/T − π/2) + R + clearance

Forensics 1

Impact angle of a trajectory impact angle = tan⁻¹(vertical drop / horizontal travel)

Graphics & Technology 1

2D Rotation (Graphics) x′ = x·cosθ − y·sinθ; y′ = x·sinθ + y·cosθ

Heights & Distances 1

Measuring a tree's height height = distance × tan θ + eye height

Marine Biology 1

Ocean depth from angled sonar depth = slant range × sin θ

Navigation & Aviation 1

Tracking an aircraft's altitude altitude = ground distance × tan θ

Oceanography 1

Tide Height Prediction h(t) = A·sin(2πt/T) + mean

Physics & Motion 1

Pendulum Period & Swing T = 2π√(L/g); θ(t) = θ₀·cos(2πt/T)

Advanced Applications 8

Euler's Formula & de Moivre e^{iθ} = cos θ + i sin θ; (r(cosθ+isinθ))ⁿ = rⁿ(cos nθ + i sin nθ)

Inverse Trigonometric Functions arcsin x ∈ [−90°,90°]; arccos x ∈ [0°,180°]; arctan x ∈ (−90°,90°)

Modelling Periodic Phenomena y = A sin(B(x − C)) + D — amplitude |A|, period 360°/B, midline D

Polar Coordinates x = r cos θ, y = r sin θ; r = √(x²+y²), θ = atan2(y, x)

Real-World Trig Problems elevation: h = d·tanθ; ladder: h = L·sinθ; wheel: y = D + R·sin(ωt)

Reciprocal Trig Functions sec θ = 1/cos θ; cosec θ = 1/sin θ; cot θ = cos θ/sin θ

Small-Angle Approximation sin x ≈ x, tan x ≈ x, cos x ≈ 1 − x²/2 (x in radians)

Wave Superposition (a sinx + b cosx) a sin x + b cos x = R sin(x + α), R = √(a²+b²), α = arctan(b/a)

Foundations 2

Intro to sin, cos, tan sin = O/H · cos = A/H · tan = O/A (SOH-CAH-TOA)

Similar triangles & ratios a₁ / a₂ = b₁ / b₂ (corresponding sides are in proportion)

Non-Right-Angle Trigonometry 5

Ambiguous case (SSA) sin B = b·sin A / a → B or 180°−B may both be valid

Area of any triangle (½·a·b·sin C) Area = ½·a·b·sin C (a, b sides; C the angle between them)

Cosine rule c² = a² + b² − 2ab·cos C

Heron's formula via the cosine rule Area = ½·a·b·sin C, with cos C = (a²+b²−c²)/(2ab) → Heron's √(s(s−a)(s−b)(s−c))

Sine rule a / sin A = b / sin B = c / sin C

Right-Angle Trigonometry 5

3D trigonometry diagonal = √(l² + w² + h²) · angle = tan⁻¹(h / √(l²+w²))

Angles of elevation & depression tan θ = height / distance

Compass bearings bearing = (90° − atan2(N, E)) mod 360°

Find a side opp = hyp·sinθ · adj = hyp·cosθ · opp = adj·tanθ

Find an angle θ = sin⁻¹(O/H) · cos⁻¹(A/H) · tan⁻¹(O/A)

Trigonometric Equations 2

General Solution sin: x=α+360°n or 180°−α+360°n; cos: x=±α+360°n; tan: x=α+180°n

Solving Trig Equations sin x = k → x = sin⁻¹k and its symmetric partners, within the domain

Trigonometric Functions & Graphs 7

Degrees & radians radians = degrees × π/180 · degrees = radians × 180/π

Period & amplitude amplitude = |A| · period = 360°/|B| = 2π/|B|

The unit circle P = (cos θ, sin θ) on the circle of radius 1

Transformations: y = A·sin(Bx + C) + D y = A·sin(Bx + C) + D

y = cos x y = cos x (amplitude 1, period 360° = 2π)

y = sin x y = sin x (amplitude 1, period 360° = 2π)

y = tan x y = tan x = sin x / cos x (period 180°, asymptotes where cos x = 0)

Trigonometric Identities 6

Compound-Angle Formulae sin(A±B)=sinA cosB ± cosA sinB; cos(A±B)=cosA cosB ∓ sinA sinB

Double-Angle Formulae sin2A=2sinAcosA; cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A; tan2A=2tanA/(1−tan²A)

Exact Trigonometric Values sin 30°=1/2, cos 30°=√3/2, sin 45°=cos 45°=√2/2, sin 60°=√3/2

Half-Angle Formulae sin(A/2)=±√((1−cosA)/2); cos(A/2)=±√((1+cosA)/2); tan(A/2)=(1−cosA)/sinA

Product-to-Sum Formulae 2sinAcosB=sin(A+B)+sin(A−B); 2cosAcosB=cos(A−B)+cos(A+B)

Pythagorean Identities sin²θ + cos²θ = 1

Chaos & Dynamics 1

Lorenz Attractor the butterfly effect in 3D

Famous Equations 2

Euler's Formula e^(iθ) = cos θ + i sin θ

Fourier Series building a square wave from sine waves

Sequences & Patterns 4

Collatz Trajectories the 3n+1 hailstone paths

Divisor Surface height = number of divisors d(n)

Factorial Spiral n! as a log-height spiral

Prime Spiral primes on a golden-angle spiral

Surfaces & Shapes 2

Morphing Surface z = f(x, y) animated over time

Pascal Terrain binomial coefficients as a 3D terrain

Probability 3

Combination Calculator nCr = n! / (r! (n − r)!)

Permutation Calculator nPr = n! / (n − r)!

Probability Calculator P = favourable outcomes / total outcomes

Statistics 5

Average (Mean) Calculator mean = (sum of values) / (count of values)

Median Calculator median = middle value of the sorted list

Mode Calculator mode = the value(s) that appear most often

Standard Deviation Calculator σ = √(Σ(x − mean)² / N) ; sample s divides by N − 1

Variance Calculator population σ² = Σ(x − mean)² / N ; sample s² divides by N − 1

∑ Maths

Algebra