§-Unit

WASpecialist Mathematics← Specialist Mathematics syllabus

Unit 3

19 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do we restrict the circular functions so that their inverses are genuine functions, and what do those inverses look like?

Define the inverse circular functions with their restricted domains and ranges and sketch their graphs

WACE Specialist Unit 3 inverse circular functions: why sine, cosine and tangent must be domain-restricted to be invertible, the principal domains and ranges of arcsin, arccos and arctan, their graphs as reflections, and exact values, with a worked example.

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How can we raise a complex number to a high integer power without expanding the binomial?

State and apply de Moivre's theorem to evaluate integer powers of complex numbers and derive trigonometric identities

WACE Specialist Unit 3 de Moivre's theorem: raising a complex number in polar form to an integer power, its proof by induction, evaluating powers quickly, and deriving multiple-angle trigonometric identities, with a worked example.

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How do we extend the real numbers to solve every polynomial equation?

Represent complex numbers in Cartesian and polar form, perform arithmetic, and apply de Moivre's theorem

WACE Specialist Unit 3 complex numbers: Cartesian and polar (modulus-argument) form, the Argand plane, arithmetic, conjugates, de Moivre's theorem and the nth roots of a complex number, with full worked examples.

8 min read →

How does plotting a complex number reveal its size and direction through modulus and argument?

Represent complex numbers on the Argand plane and find modulus and argument, converting to polar form

WACE Specialist Unit 3 complex plane: plotting on the Argand diagram, the modulus as distance from the origin, the argument and principal argument in the interval negative pi to pi, quadrant checks, and conversion to polar form, with a worked example.

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Given the graph of y equals f of x, how do we sketch the graph of its reciprocal one over f of x?

Sketch the reciprocal of a function, relating zeros to asymptotes and turning points to turning points

WACE Specialist Unit 3 reciprocal graphs: how zeros of f become vertical asymptotes of one over f, where the reciprocal is large or small, sign preservation, fixed points at plus and minus one, and turning point behaviour, with a worked example.

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How do transformations and structure determine the shape of a graph?

Sketch rational functions, reciprocal and modulus graphs, and use transformations and asymptotic behaviour

WACE Specialist Unit 3 functions and graphs: rational functions, vertical and horizontal asymptotes, the reciprocal of a graph, modulus functions, and curve sketching from intercepts, asymptotes and turning points.

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How do the zeros of the numerator and denominator control the shape of a rational function's graph?

Sketch graphs of rational functions, identifying intercepts, vertical, horizontal and oblique asymptotes

WACE Specialist Unit 3 rational functions: vertical asymptotes from denominator zeros, horizontal and oblique asymptotes from degree comparison, intercepts, holes from common factors, and sign analysis for sketching, with a worked example.

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How do the two absolute value transformations reshape a graph, and how do we solve modulus equations?

Sketch graphs of the modulus functions y equals the absolute value of f of x and y equals f of the absolute value of x, and solve modulus equations

WACE Specialist Unit 3 modulus functions: reflecting the negative part upward for y equals modulus of f, mirroring the right side for y equals f of modulus x, and solving absolute value equations and inequalities by cases, with a worked example.

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How do equations and inequalities in z carve out lines, circles and regions in the Argand plane?

Describe and sketch subsets of the complex plane defined by equations and inequalities in modulus and argument

WACE Specialist Unit 3 loci in the complex plane: circles from modulus conditions, perpendicular bisectors from equal distances, rays from argument conditions, and shaded regions from inequalities, translated to Cartesian form, with a worked example.

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How do we differentiate and integrate the harder functions of Specialist?

Differentiate inverse trig functions, use implicit differentiation, and integrate rational functions, partial fractions and trig forms

WACE Specialist Unit 3 further calculus: derivatives of inverse trigonometric functions, implicit differentiation, related rates, and integration by recognition, substitution and the standard inverse-trig and logarithmic forms, with full worked examples.

9 min read →

How do we describe the position of a moving particle by a vector function and differentiate it to get velocity and acceleration?

Use vector functions of time and differentiate them to find velocity, speed and acceleration along a path

WACE Specialist Unit 3 vector calculus: position vectors as functions of time, componentwise differentiation to velocity and acceleration, speed as the magnitude of velocity, the cartesian path equation, and motion problems, with a worked example.

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How do vectors describe direction, length and angle in space?

Use 3D vectors with the dot product and cross product to find lengths, angles, projections and areas

WACE Specialist Unit 3 three-dimensional vectors: components, magnitude, unit vectors, the scalar (dot) product for angles and projections, and the vector (cross) product for perpendiculars and areas.

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How does admitting complex roots let every polynomial factor completely, and how do real polynomials behave?

Factorise polynomials over the complex numbers using the conjugate root theorem and the fundamental theorem of algebra

WACE Specialist Unit 3 factorisation over C: the fundamental theorem of algebra, the conjugate root theorem for real polynomials, pairing complex roots into real quadratic factors, and finding all roots from one, with a worked example.

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How does the dot product measure the angle between two three-dimensional vectors and project one onto another?

Compute the scalar (dot) product of vectors in three dimensions and use it for angles, perpendicularity and projections

WACE Specialist Unit 3 scalar product: the component and geometric definitions of the dot product, the angle formula, the perpendicularity test, and scalar and vector projections, with a worked example in three dimensions.

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How do we build a vector perpendicular to two given vectors, and why does its length measure area?

Compute the vector (cross) product in three dimensions and use it to find perpendicular vectors and areas

WACE Specialist Unit 3 cross product: the determinant definition, the right-hand rule, why the result is perpendicular to both vectors, the magnitude as parallelogram area, and the triangle area, with a worked example.

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Why does every nonzero complex number have exactly n distinct nth roots, evenly spaced on a circle?

Find the nth roots of a complex number and the nth roots of unity, and represent them on the Argand plane

WACE Specialist Unit 3 roots of complex numbers: solving z to the n equals w with the general root formula, the nth roots of unity, equal spacing on a circle of radius the nth root of the modulus, and their sum, with a worked example.

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Why does multiplying complex numbers rotate and scale, and how does polar form make this obvious?

Multiply and divide complex numbers in polar form, interpreting the result as rotation and scaling

WACE Specialist Unit 3 polar form arithmetic: multiplying moduli and adding arguments, dividing moduli and subtracting arguments, the geometric interpretation as rotation and dilation, and the conjugate in polar form, with a worked example.

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How do the four arithmetic operations work once we admit the number i with i squared equal to negative one?

Perform addition, subtraction, multiplication and division of complex numbers in Cartesian form using the conjugate

WACE Specialist Unit 3 complex arithmetic in Cartesian form: adding, subtracting, multiplying and dividing $z = x + iy$ , the conjugate, the realising trick for division, and powers of $i$ , with full worked SCSA-style calculations.

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How does a vector equation with a parameter trace out a curve, and how do we convert it to cartesian form?

Write vector and parametric equations of curves and convert between vector, parametric and cartesian forms

WACE Specialist Unit 3 vector and cartesian equations of curves: parametrising a path with a vector equation, reading off component equations, eliminating the parameter to get a cartesian relation, and recognising standard curves, with a worked example.

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