NSW Syllabuses

# Mathematics Extension 2 Stage 6 - Year 12 Extension 2 - Complex Numbers MEX-N2 Using Complex Numbers

## Outcomes

#### A student:

• MEX12-1

understands and uses different representations of numbers and functions to model, prove results and find solutions to problems in a variety of contexts

• MEX12-4

uses the relationship between algebraic and geometric representations of complex numbers and complex number techniques to prove results, model and solve problems

• MEX12-7

applies various mathematical techniques and concepts to model and solve structured, unstructured and multi-step problems

• MEX12-8

communicates and justifies abstract ideas and relationships using appropriate language, notation and logical argument

## Subtopic Focus

The principal focus of this subtopic is to develop and apply complex number knowledge to situations involving trigonometric identities, powers and vector representations in a complex number plane.

Students develop appreciation of the interconnectedness of complex numbers across various topics of mathematics and their applications in real life.

## Content

• N2.1: Solving equations with complex numbers
• Students:
• prove De Moivre’s theorem for integral powers using proof by induction (ACMSM083)
• solve power equations (ACMSM082)
• use De Moivre's theorem to derive trigonometric identities such as $$\cos 3 \theta = 4 \cos^{3} \theta - 3 \cos \theta$$
• solve quadratic equations of the form $$ax^2 + bx+c = 0$$, where $$a$$, $$b$$, $$c$$ are complex numbers (ACMSM072, ACMSM074, ACMSM075)
• define and use conjugate roots for polynomials with real coefficients, and show that roots of polynomials with real coefficients occur as conjugate pairs (ACMSM090)
• solve problems involving real polynomials with conjugate roots
• N2.2: Geometrical implications of complex numbers
• Students:
• given the points representing $$z_1$$ and $$z_2$$, find the position of the point representing $$z_1+z_2$$ and $$z_1-z_2$$
• describe the vector representing $$z_1+z_2$$ or $$z_1-z_2$$ as corresponding to the relevant diagonal of a parallelogram with vectors representing $$z_1$$ and $$z_2$$ as adjacent sides
• examine and use the geometric interpretation of multiplying complex numbers, including rotation and scaling in the Argand plane (ACMSM085)
• explain why multiplying by $$i$$ gives a $$\frac{\pi}{2}$$ rotation anticlockwise about $$(0,0)$$, and multiplying by $$-1$$ gives a rotation of $$\pi$$ in the Argand plane
• recognise the geometrical relationship between the points representing a complex number $$z=a+ib$$, and the points representing $$\bar z$$, $$cz$$ (where $$c$$ is real) and $$iz$$ (ACMSM073)
• determine and examine the $$n$$th roots of unity and their location on the unit circle (ACMSM087)
• determine and examine the $$n$$th roots of complex numbers and their location in the Argand plane (ACMSM088)
• solve problems using $$n$$th roots of complex numbers
• identify subsets of the Argand plane determined by relations, for example, $$\left|z-3i\right|\le 4$$, $$\frac{\pi}{4} \le Arg(z) \le \frac{3\pi}{4}$$, $$Re(z) > Im(z)$$ and $$\left|z-1\right| = 2\left|z-i\right|$$ (ACMSM086)