Trigonometry

Trigonometry is the maths of triangles: it lets you find a length or an angle you can't measure directly, like the height of a cliff or the width of a river. In this topic you'll revise Pythagoras' theorem, discover by measuring that every angle in a right-angled triangle has its own fixed ratios, and then use those ratios, sine, cosine and tangent, to find missing sides and angles in real problems.

What you'll learn

• 8.01 Pythagoras' theorem: finding the hypotenuse and the shorter sides, exact answers, and checking whether a triangle is right-angled.
• Investigation: the triangle inequality. Test which lengths make a triangle with a ruler and compass, then write and refine an algorithm that decides.
• Investigation: ratios in triangles. Measure similar right-angled triangles at true size and discover that their side ratios never change.
• 8.02 Trigonometric ratios: naming the hypotenuse, opposite and adjacent sides, and writing $\sin \theta$, $\cos \theta$ and $\tan \theta$ (SOH CAH TOA).
• 8.03 Side lengths: using sin, cos and tan to find a missing side, including when the unknown is on the bottom of the fraction.
• 8.04 Angles: using the inverse ratios to find a missing angle.
• 8.05 Applications of trigonometry: angles of elevation and depression, true bearings, and problems with more than one triangle.

Which level should I pick?

Every worksheet comes in three levels. Each one covers the same skills.

• Level 1 · Warm-up: simpler triangles and more help along the way. A good place to start.
• Level 2 · Core: the main worksheet for the lesson.
• Level 3 · Challenge: multi-step problems and harder reasoning, for when the Core level feels easy.

Not sure? Start with Level 2. If the first few questions feel tricky, drop to Level 1. If you finish quickly and get them right, try Level 3. Download the worksheet, print it, and write your answers in the boxes. You'll need a scientific calculator (in degree mode) for most of this topic, and a ruler, compass and protractor for the investigations.

Practice tests

When you've finished the lessons, try the three practice tests. Each one is like the real topic test: 45 minutes, 40 marks, a section without a calculator and a section with one. Write your working under each question, then copy your final answers onto the answer sheet on the last page, just like you will in the real test.