This is Chapter 7 – Trigonometric Functions for the Year 11 Mathematical Methods course.

This chapter builds directly on the unit circle and develops trigonometry from a collection of triangle-based techniques into a powerful way of modelling repeating behaviour. Students begin to treat sine, cosine and tangent as functions, examining how their equations determine the shape, position and behaviour of their graphs.

We first explore the meaning of periodic behaviour and how it appears in situations that repeat over regular intervals. The sine and cosine functions are then studied in detail, with students learning how amplitude, period, horizontal shifts and vertical translations affect each graph. A major focus is placed on moving confidently between an equation, its graph and the real situation being modelled.

The tangent function introduces some additional complexity because its graph contains asymptotes and is not defined for every input. Students will need to pay close attention to domains, periods and the location of vertical asymptotes when sketching graphs or solving equations.

Trigonometric equations also require a more systematic approach than many earlier algebraic equations. Students must identify all solutions within a stated interval, use unit-circle relationships accurately and recognise when multiple answers are required. Angle sum and difference identities then extend students’ algebraic fluency, allowing exact values and expressions to be developed from known angles.

These functions are widely used to model tides, sound waves, seasonal patterns, alternating electrical signals, circular motion and vibration. They also form an essential foundation for Year 12 calculus, where students differentiate and integrate trigonometric functions and apply them to rates of change and motion.

Subchapters:

• 7A: Periodic behaviour
• 7B: The sine function
• 7C: The cosine function
• 7D: The tangent function
  
• 7E: Trigonometric equations
  
• 7F: Using trigonometric functions
• 7G: Angle sum and difference identities

By the end of this chapter, students will be able to graph, transform and apply trigonometric functions, solve trigonometric equations and use identities to analyse periodic relationships with confidence.