GCSE Maths: Pythagoras and Trigonometry
Pythagoras' theorem and trigonometry are the highest-value geometry skills at GCSE: they appear on every paper, they anchor multi-step problems, and at Higher tier they extend into the sine and cosine rules for any triangle. This quiz builds the full progression, with every triangle described completely in words so the method choice is always yours.
The Pythagoras questions cover both directions — finding a hypotenuse from the two shorter sides (the 6-8-10 case) and finding a shorter side from the hypotenuse (15 and 9 giving 12) — the second being where subtraction errors traditionally appear. A method-choice question then asks when trigonometry is needed instead of Pythagoras, testing the decision the specification calls "knowing which tool fits the information given".
The core trigonometry questions apply sine, cosine and tangent in right-angled triangles: using sin 30° = 0.5 to find a hypotenuse from an opposite side, reading cos θ from adjacent and hypotenuse, and recognising that equal opposite and adjacent sides force θ = 45°. The exact trigonometric values for 0°, 30°, 45°, 60° and 90° — which both tiers must know without a calculator — are tested directly.
Higher tier content completes the set: the statements of the sine rule and cosine rule, and a strategy question about which rule finds an angle when all three sides are known. The conceptual questions about the tangent ratio (opposite over adjacent, and tan = sin/cos) round out understanding beyond mechanical substitution. Explanations show the full rearrangement in every case, and distractors encode the classic confusions — swapped ratios, added instead of subtracted squares, and misremembered exact values.
- Apply Pythagoras' theorem to find the hypotenuse or a shorter side
- Choose correctly between Pythagoras and trigonometry from the information given
- Use sine, cosine and tangent to find sides and angles in right-angled triangles
- Recall exact trigonometric values for 0°, 30°, 45°, 60° and 90°
- State and select between the sine rule and cosine rule for non-right-angled triangles (Higher)
Topic scope follows the Geometry and measures strand of the Department for Education's GCSE mathematics subject content: Pythagoras' theorem, the trigonometric ratios in right-angled triangles, exact trig values required at both tiers, and (Higher) the sine rule, cosine rule and area formula for any triangle.