This worksheet builds conceptual understanding of trigonometry, teaching students to explain — using similar triangles — why the sine, cosine and tangent ratios depend only on an angle's size. The worksheet is split into warm-up questions, standard questions and then extension questions to test you.
The concepts it covers:
- Two right-angled triangles that share the same acute angle are similar (aa similarity — both have a right angle plus the shared angle, so the third angle must also match).
- In similar triangles, corresponding sides are proportional — so any ratio of two sides (like opposite÷hypotenuse) gives the same value, no matter how big or small the triangle is.
- This is exactly why sin(θ), cos(θ) and tan(θ) are well-defined: they depend only on the angle θ, not on which particular right triangle you measured them in.
- A 30-60-90 triangle always has side ratio 1 : √3 : 2 (short leg : long leg : hypotenuse), no matter its size. A 45-45-90 triangle always has ratio 1 : 1 : √2.
- This constancy is what allows trigonometry to make predictions — measure a ratio in one (small, convenient) triangle, and apply it to a different (large, hard-to-measure) triangle with the same angle.
What your child will practise:
- Calculating a side ratio in a right triangle, and compare it to the same ratio in a similar (scaled) triangle.
- Explaining why two right triangles sharing an acute angle must be similar.
- Using a known ratio from one right triangle to predict a missing side in a similar triangle with the same angle.
- Working with exact surd values (√2, √3) in fixed-ratio triangles (30-60-90, 45-45-90).
- Connecting the idea of a constant ratio to real-world indirect-measurement problems.
Every section opens with a worked example, and the download includes a full answer key with step-by-step solutions and teaching notes on the mistakes students most commonly make.
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