Grade 12 Technical Mathematics – Exam Review Vocabulary

Flashcards summarising essential formulas, theorems and definitions appearing in the 2021 Grade 12 Technical Mathematics Paper 2 preparatory exam.

Distance Formula

For points (x₁,y₁) and (x₂,y₂), the length of the segment joining them is d = √[(x₂ – x₁)² + (y₂ – y₁)²].

Midpoint Formula

The midpoint of the segment joining (x₁,y₁) and (x₂,y₂) is ( (x₁+x₂)/2 , (y₁+y₂)/2 ).

Gradient (Slope)

For a line through (x₁,y₁) and (x₂,y₂), the gradient m = (y₂ – y₁)/(x₂ – x₁).

Parallel Lines (Cartesian)

Two non-vertical lines are parallel when they have equal gradients: m₁ = m₂.

Perpendicular Lines (Cartesian)

Two non-vertical lines are perpendicular when their gradients satisfy m₁·m₂ = –1.

Straight-line Equation (Slope–Intercept)

y = mx + c, where m is the gradient and c the y-intercept.

Straight-line Equation (Point–Slope)

y – y₁ = m(x – x₁) for a line of gradient m through (x₁,y₁).

Parallelogram Test (Coordinate)

A quadrilateral is a parallelogram if both pairs of opposite sides are parallel (or equal).

Tangent to a Circle

A straight line that touches a circle at exactly one point without cutting it.

Radius–Tangent Relationship

The radius drawn to the point of tangency is perpendicular to the tangent line.

Circle Equation (Centre at Origin)

x² + y² = r², where r is the radius.

Angle of Elevation

The acute angle measured upward from the horizontal to an object.

Period of a Trig Function

The smallest positive x-interval over which the graph repeats (360° for sin and cos).

Amplitude of a Trig Function

Half the vertical distance between the maximum and minimum values of the function.

Cosine Rule

For ∆ABC, a² = b² + c² – 2bc·cos A.

Sine Rule

a/sin A = b/sin B = c/sin C for any ∆ABC.

Area of a Triangle (Trig)

Area = ½ab·sin C where C is the included angle between sides a and b.

Pythagorean Identity

sin²θ + cos²θ = 1 for all real θ.

Reciprocal Trig Ratios

cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.

Secant (Trig)

sec θ is the reciprocal of cosine: sec θ = 1/cos θ.

Cosecant (Trig)

cosec θ is the reciprocal of sine: 1/sin θ.

Cotangent (Trig)

cot θ is the reciprocal of tangent: cos θ/sin θ.

Co-function Identities

sin(90°–θ)=cos θ and cos(90°–θ)=sin θ.

Period of cos x

360° (or 2π rad).

Period of sin x

360° (or 2π rad).

Arc Length

In radians, s = rθ, where r = radius and θ = central angle.

Sector Area

Area = ½r²θ, with θ in radians.

Angular Velocity

ω = 2πn rad s⁻¹, where n is the rotation frequency (rev s⁻¹).

Circumferential (Linear) Velocity

v = πDn, where D = diameter and n = rev s⁻¹.

Rotation Frequency

Number of complete revolutions per second (Hz or rev s⁻¹).

Parallel-Side Theorem (Triangles)

A line parallel to one side of a triangle divides the other two sides proportionally.

Mid-ordinate Rule

Approximate area: A ≈ a(m₁ + m₂ + … + mₙ), where a = equal part width and mᵢ are mid-ordinates.

Surface Area of a Cone

A = πr² + πrs (base + curved surface), where s = slant height.

Surface Area of a Cylinder

A = 2πr² + 2πrh (two bases plus curved surface).

Volume of a Cone

V = (1/3)πr²h.

Volume of a Cylinder

V = πr²h.

Chord Perpendicular Theorem

The line from the centre perpendicular to a chord bisects the chord.

Exterior Angle of Cyclic Quadrilateral

Equals the interior opposite angle.

Equal Angles Same Chord

Angles subtended by the same chord on the same side of the chord are equal.

Tangent–Chord Angle Theorem

The angle between a tangent and a chord equals the angle in the opposite arc.

Diameter Perpendicular to Chord

A diameter that is perpendicular to a chord passes through the chord’s midpoint and bisects the arc.

Intercept of Tangent Line

For a tangent with x- and y-intercepts D and C, the line equation can be written in intercept form x/a + y/b = 1.

Standard Ellipse Equation

x²/a² + y²/b² = 1, with a, b the semi-axes.

Parallelogram Opposite Sides

In a parallelogram, opposite sides are parallel and equal in length.