Pythagoras' theorem and trigonometry
In a right-angled triangle, Pythagoras links the three sides and trigonometry links sides with angles. The skill is choosing which one a question needs.
Part 1 of 3: Learn it
In short
- Pythagoras: a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle).
- SOHCAHTOA: sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent.
- No angle in the question or answer? Use Pythagoras. An angle involved? Use trigonometry.
Where this is in your specification
Spec points: DfE G6, G20 and G21 (sine and cosine rules, G22, are Higher tier)
| Board | Topic: Pythagoras and trigonometry |
|---|---|
| DfE content | G6, G20, G21, G22, G23 |
| AQA 8300 | Geometry and measures |
| Edexcel 1MA1 | Geometry and measures |
| OCR J560 | 10 Mensuration |
Pythagoras' theorem
- Finding the hypotenuse: square the two shorter sides, add, then square-root.
- Finding a shorter side: square the hypotenuse, subtract the square of the other side, then square-root.
- Sense check: the hypotenuse is always the longest side.
Labelling for trigonometry
Mark the angle you are using. The hypotenuse is opposite the right angle, the opposite is across from your angle, and the adjacent is next to your angle (the side that is not the hypotenuse).
Pick the ratio that uses the two sides you know or want. To find an angle, use the inverse button: θ = tan⁻¹(O ÷ A), and so on.
Exact values
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan | 0 | √3/3 | 1 | √3 | not defined |
Non-calculator papers can ask for these, so learn them or learn to rebuild them from two triangles: half an equilateral triangle of side 2, and a right-angled isosceles triangle with shorter sides of 1.
Beyond right angles (Higher)
For triangles without a right angle, the sine rule (a ÷ sin A = b ÷ sin B) works when you know a side and its opposite angle; the cosine rule (c² = a² + b² − 2ab cos C) works with three sides, or two sides and the angle between them. Area = ½ab sin C.
A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.
Show the answer
√(36 + 64) = √100 = 10 cm.
Part 2 of 3: See it worked
Worked examples
Example 1
A ladder 10 m long leans against a wall with its foot 6 m from the wall. How high up the wall does it reach?
- The ladder is the hypotenuse, so find a shorter side
- Square both known sides: 10 × 10 = 100 and 6 × 6 = 36
- Subtract, because h is a shorter side: 100 − 36 = 64
- Square-root: h = √64 = 8
Answer: 8 m.
Example 2
In a right-angled triangle the side opposite angle θ is 7 cm and the side adjacent to it is 9 cm. Find θ.
- Opposite and adjacent, so use tan
- tan θ = 7 ÷ 9 = 0.7778
- θ = tan⁻¹(0.7778) = 37.87°
Answer: θ = 37.9° (1 d.p.).
Common mistakes
- Adding the squares when you need a shorter side.
- Forgetting the square root at the end of Pythagoras.
- Calculator in radians or gradians: check for a small D or DEG on the screen.
- Labelling opposite and adjacent from the right angle instead of from the angle you are using.
The hypotenuse is 10 cm and the angle is 35°. Find the side opposite the angle.
Show the answer
10 × sin 35° = 5.74 cm.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.
√(36 + 64) = √100 = 10 cm.
The hypotenuse is 10 cm and the angle is 35°. Find the side opposite the angle.
10 × sin 35° = 5.74 cm.
What is cos 60° exactly?
1/2.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Estos apuntes están en inglés porque siguen los programas de examen del Reino Unido.
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses