This is Chapter 6 – Non-Right Angled Triangle Trigonometry for the Year 11 Mathematical Methods course.

This chapter extends trigonometry beyond right-angled triangles and develops methods for solving triangles of any shape. Students must learn to interpret the information provided, choose an appropriate relationship and check whether the resulting solution makes sense within the geometry of the problem.

We begin with the area formula for a triangle involving two sides and the included angle. This provides an alternative to base–height calculations and reinforces the connection between trigonometric ratios and geometric measurement.

The cosine rule is then introduced for situations where the standard right-angle ratios do not apply. Students use it to determine unknown sides or angles and must recognise which arrangement of known information makes the rule appropriate. Rearranging the formula accurately is an important algebraic skill.

The sine rule complements this by linking corresponding sides and opposite angles. Particular care is required in ambiguous cases, where the same information can sometimes produce more than one valid triangle. Students must consider all possible solutions and decide which are mathematically or contextually acceptable. The ambiguous case is particularly important because a calculator may return only one angle initially, so students must use geometric reasoning to determine whether a second triangle is possible.

Problem-solving questions bring the techniques together and often require labelled diagrams, multiple stages and careful interpretation of units. Non-right angled trigonometry is used in surveying, navigation, engineering and physics and also strengthens the geometric reasoning required for later trigonometric functions and modelling.

Subchapters:

• 6A: Areas of triangles
  
• 6B: The cosine rule
• 6C: The sine rule
• 6D: Problem solving

By the end of this chapter, students will be able to select and apply area, sine and cosine rules to solve non-right angled triangles and justify solutions in unfamiliar contexts.