GCSE Maths Trigonometry (SOH CAH TOA) Notes
Trigonometry uses sin, cos and tan to link the sides and angles of right-angled triangles. SOH CAH TOA tells you which ratio to pick.
Key concepts
- Opposite, adjacent, hypotenuse
- Label sides relative to the angle: opposite faces it, adjacent is next to it, hypotenuse is longest.
- SOH CAH TOA
- sin = O/H, cos = A/H, tan = O/A. Choose the ratio using the two sides involved.
- Finding an angle
- Use the inverse functions sin⁻¹, cos⁻¹, tan⁻¹ on your calculator.
Key formulas
Sine ratio
sin θ = opposite ÷ hypotenuse
Cosine ratio
cos θ = adjacent ÷ hypotenuse
Tangent ratio
tan θ = opposite ÷ adjacent
Worked example
A right-angled triangle has an angle of 30° and a hypotenuse of 12 cm. Find the opposite side.
- 1Opposite and hypotenuse are involved, so use sine: sin 30° = O / 12.
- 2Rearrange: O = 12 × sin 30°.
- 3Since sin 30° = 0.5, O = 12 × 0.5 = 6.
Answer: 6 cm
Common mistakes
✗ Choosing the wrong ratio.
✓ Label the sides first, then pick the ratio that uses the two you know or want (SOH CAH TOA).
✗ Calculator in the wrong mode.
✓ Set it to degrees (DEG) for GCSE, not radians.
✗ Forgetting the inverse when finding an angle.
✓ Use sin⁻¹/cos⁻¹/tan⁻¹ to turn a ratio back into an angle.
Quick check
Try these, then reveal the answers.
Which ratio uses opposite and adjacent?Reveal answer
tan
sin 90° = ?Reveal answer
1
To find an angle from tan, which button?Reveal answer
tan⁻¹ (inverse tan).
Exam tips
- Write SOH CAH TOA at the top of the question to choose the ratio quickly.
- Check your calculator is in degree mode before you start.
- Round only at the end and give angles to 1 decimal place unless told otherwise.
Related practice
Practise this topic
Exam-style questions with instant Study Coach feedback.
Flashcards
Quick-fire recall to lock in the key facts.
Worksheets
Printable practice you can work through offline.
Past papers
Full papers with examiner-grade marking.
Study Coach
Ask for a step-by-step explanation any time.
All Mathematics
Every Mathematics topic across every exam board.
Related topics
Key terms explained
Short, exam-ready definitions with a formula, a worked example and an examiner tip.
Frequently asked questions
How do I choose between sin, cos and tan?+
Label the sides you have or need relative to the angle, then pick the ratio that uses those sides.
How do I find a missing angle?+
Work out the ratio of the two sides, then apply the inverse function (sin⁻¹, cos⁻¹ or tan⁻¹).
Create a free account to practise trigonometry (soh cah toa) with Study Coach feedback
Try 5 questions now, get instant examiner-grade marking and ask the Study Coach to explain anything. Free during our early launch period — no credit card required.
Related on MathsGradeUp
Keep studying
- Mathematics courses & specifications
Every board and level we cover for mathematics.
- Formula sheets
Every GCSE, IGCSE and A Level formula, searchable and explained.
- Maths & science glossary
Plain-English definitions with a formula, worked example and examiner tip.
- Free maths calculators & solvers
Step-by-step solvers for the calculations that come up in every paper.
- Past paper guides
Paper structure, timings and topic weightings board by board.
- Grade boundaries
Historic boundaries and what each grade typically needs.
- Browse subjects
Specification-mapped courses for every board and level.
Practise it
- Predicted papers
GCSE and A Level predicted papers with full mark schemes.
- Question bank
Filter thousands of exam-board questions by topic and difficulty.
- Past papers workspace
Write on real papers and get every line marked.