This is Chapter 4 – Trigonometric Functions for the Year 11 Specialist Mathematics course.

This chapter builds on the trigonometry studied in Mathematical Methods and takes a much deeper look at the algebraic and graphical behaviour of trigonometric functions. Students move beyond the standard sine, cosine and tangent graphs and begin working with more general equations, transformations, identities and function relationships.

We begin by examining general sine, cosine and tangent functions, focusing on how features such as amplitude, period, horizontal shifts and vertical translations are determined from an equation. Students will need to move confidently between equations and graphs, recognise the effects of different parameters and interpret these features within modelling contexts.

Reciprocal trigonometric functions introduce additional graphical behaviour, including restricted domains, vertical asymptotes and repeated branches. Students then develop more systematic methods for solving trigonometric equations, where it is essential to identify every valid solution within a stated interval rather than stopping after the first answer.

The chapter also places a strong emphasis on trigonometric identities. Students work with fundamental relationships, double-angle identities, and angle sum and difference identities before applying these ideas to equations written in quadratic form. These sections require accurate algebra, strong pattern recognition and the ability to select an appropriate identity without always being told which one to use.

Trigonometric functions are widely used to describe waves, oscillations, rotation and other forms of periodic motion. They are especially important in physics, engineering, sound analysis, electronics and signal processing. The skills developed here also provide essential preparation for the trigonometric calculus and modelling encountered in Stage 2 Mathematics.

Subchapters:

• 4A: General trigonometric functions – Sine

• 4B: General trigonometric functions – Cosine
• 4C: The general tangent function
• 4D: Reciprocal trigonometric functions
• 4E: Solving trigonometric equations
• 4F: Modelling using trigonometric functions
• 4G: Trigonometric relationships
• 4H: Double angle identities
• 4I: Angle sum and difference identities
• 4J: Trigonometric equations in quadratic form

By the end of this chapter, students will be able to graph, transform, model and solve with a wide range of trigonometric functions while using identities accurately and strategically.