Chapter 20: Pythagoras' theorem and trigonometry

The chapter problem

Cross the garden. A rectangular garden is $9$ m long and $12$ m wide. A path runs straight from one corner to the opposite corner. Estimate the path length before calculating. There is no mark for the estimate; use it to decide whether your final answer makes sense.

By the end of the chapter, you will find missing lengths and angles in right-angled and other triangles, including triangles in three dimensions.

By the end of this chapter you will be able to

• use Pythagoras' theorem and trigonometric ratios in right-angled triangles;
• model elevation and depression angles;
• use exact trigonometric values and solve 3D triangle problems;
• choose the sine rule, cosine rule, or triangle area formula when appropriate.

Where it is used

Triangle methods can help estimate a direct distance, a roof slope, or a height that cannot be measured directly.

Your turn to connect it

Where might you or someone you know need to find a distance or height without measuring it directly? Write one example and say what information you could measure.

Roadmap

1. 20.1 Pythagoras' theorem
2. 20.2 Trigonometric ratios (SOHCAHTOA)
3. 20.3 Angles of elevation and depression
4. 20.4 Exact trigonometric values
5. 20.5 3D Pythagoras and trigonometry
6. 20.6 The sine rule
7. 20.7 The cosine rule
8. 20.8 The area of any triangle
9. Review

Each section has Learn, Practise with help, and Practise pages. The Review returns to the garden problem and mixes methods.