Cosine rule

“When there's no right angle, the cosine rule steps in.”

c² = a² + b² − 2ab·cosC generalises Pythagoras to any triangle.

c²=a²+b²−2ab·cosC

Memory trick: Pythagoras with a correction term.

Distance formula

“It's Pythagoras wearing coordinate-geometry clothes.”

The distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²].

d=√(Δx²+Δy²)

Memory trick: Difference-of-coordinates, squared, rooted.

Slope of a line

“Every rate you plot — speed, cost, growth — is a slope.”

Slope m = (y₂−y₁)/(x₂−x₁) = tanθ measures steepness.

m=Δy/Δx

Memory trick: Rise over run.

Parallel and perpendicular lines

“Perpendicular slopes are negative reciprocals of each other.”

Parallel lines share a slope; perpendicular slopes multiply to −1.

Memory trick: ⊥ → m₁m₂ = −1.

Equation of a circle

“Every point equidistant from the centre — written as an equation.”

A circle of centre (h,k) and radius r is (x−h)² + (y−k)² = r².

(x−h)²+(y−k)²=r²

Memory trick: Centre-radius form.

Parabola

“Satellite dishes are parabolas because they focus all signals to one point.”

The set of points equidistant from a focus and a directrix; y² = 4ax opens rightward.

y²=4ax

Memory trick: Eccentricity exactly 1.

Eccentricity of conics

“A single number e morphs a circle into an ellipse, parabola, then hyperbola.”

Circle e=0, ellipse 0<e<1, parabola e=1, hyperbola e>1.

Memory trick: e climbs, the curve opens up.

Limit

“Mathematics' way of getting infinitely close without touching.”

The value a function approaches as its input nears a point, even if undefined there.

Memory trick: Limit ≠ value at a discontinuity.

Continuity

“You can draw a continuous curve without lifting your pen.”

A function is continuous at a point if its limit there equals its value — no breaks.

Memory trick: Limit = value → continuous.

Standard limit sinx/x

“The tiny-angle secret behind much of calculus and physics.”

lim(x→0) sin x / x = 1 (x in radians).

lim(x→0)sinx/x=1

Memory trick: Only in radians.

Derivative

“Your speedometer shows a live derivative of distance.”

The instantaneous rate of change — the slope of the tangent at a point.

dy/dx

Memory trick: Slope of the tangent.

Power rule

“The workhorse of differentiation — bring the power down, drop it by one.”

d/dx(x^n) = n·x^(n−1).

d/dx(xⁿ)=nx^(n−1)

Memory trick: Drop the power, subtract one.

Product rule

“The derivative of a product is NOT the product of derivatives.”

d/dx(uv) = u'v + uv'.

(uv)'=u'v+uv'

Memory trick: First×d(second) + second×d(first).

Chain rule

“For sin(2x) it's cos(2x)×2 — never forget the ×2.”

Differentiate the outer function, then multiply by the inner derivative.

dy/dx=dy/du·du/dx

Memory trick: Outer' × inner'.

Maxima and minima

“Businesses use this to squeeze out maximum profit or minimum cost.”

At a maximum or minimum, f'(x)=0; use f''(x) to tell which (f''<0 max, f''>0 min).

Memory trick: f'=0 is only a candidate — check f''.

Increasing and decreasing

“The sign of the derivative tells you which way the curve is heading.”

Where f'(x) > 0 a function rises; where f'(x) < 0 it falls.

Memory trick: f'>0 up, f'<0 down.

Integration as antiderivative

“Undoing a derivative always loses the constant — hence +C.”

Integration reverses differentiation; indefinite integrals carry a +C.

∫xⁿ dx=x^(n+1)/(n+1)+C

Memory trick: Never forget +C.

Definite integral as area

“Adding infinitely many thin strips gives the exact area of a curved region.”

The definite integral gives the (signed) area under a curve between two limits.

Memory trick: Area below axis counts negative.

Fundamental theorem of calculus

“It's the bridge that ties the two halves of calculus together.”

Differentiation and integration are inverse operations.

Memory trick: Integrate then differentiate → back to start.

Differential equation

“Population growth, cooling coffee and radioactive decay are all differential equations.”

An equation involving a function and its derivatives.

Memory trick: Relates a function to its rate of change.

Dot product

“The dot product measures how much two vectors point the same way.”

a·b = |a||b|cosθ gives a scalar; it's zero when vectors are perpendicular.

a·b=|a||b|cosθ

Memory trick: Dot → scalar; zero → perpendicular.

Cross product

“The cross product points out of the plane, right-hand-rule style.”

a×b is a vector perpendicular to both, with magnitude |a||b|sinθ.

|a×b|=|a||b|sinθ

Memory trick: Cross → vector; zero → parallel.

Matrix

“Computer graphics rotate and scale every image using matrices.”

A rectangular array of numbers used to store and transform data.

Memory trick: Rows × columns.

Matrix multiplication is non-commutative

“Order matters: rotating then scaling differs from scaling then rotating.”

In general AB ≠ BA for matrices.

Memory trick: AB ≠ BA.