Curriculum fit: GCSE Maths Geometry and measures (Foundation and Higher). US: Geometry (Right Triangles and Trigonometry).
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What Is It?
Trigonometry links the angles and sides of right-angled triangles. For a chosen angle θ, the sides are named the hypotenuse (longest side), the opposite (across from θ) and the adjacent (next to θ).
The three ratios sine, cosine and tangent let you find a missing side or a missing angle. The memory aid is SOH CAH TOA.
Key Formulas and Ideas
- sin θ = opposite ÷ hypotenuse (SOH).
- cos θ = adjacent ÷ hypotenuse (CAH).
- tan θ = opposite ÷ adjacent (TOA).
- To find an angle, use the inverse function: θ = sin⁻¹, cos⁻¹ or tan⁻¹ of the ratio.
Worked Examples
Find a Side
A triangle has hypotenuse 10 cm and angle 30°. Find the opposite side.
- Opposite and hypotenuse means sine.
- opposite = 10 × sin 30° = 10 × 0.5 = 5.
✓ Answer 5 cm
Find an Angle
The opposite side is 3 and the adjacent is 4. Find θ.
- Opposite and adjacent means tangent: tan θ = 3 ÷ 4 = 0.75.
- θ = tan⁻¹(0.75) ≈ 36.9°.
✓ Answer 36.9°
Use Cosine
The hypotenuse is 8 cm and the angle is 60°. Find the adjacent side.
- Adjacent and hypotenuse means cosine.
- adjacent = 8 × cos 60° = 8 × 0.5 = 4.
✓ Answer 4 cm
Common Mistakes to Avoid
- Labelling the opposite and adjacent sides from the wrong angle.
- Having the calculator in radians instead of degrees.
- Using the wrong ratio for the sides given.
- Using SOH CAH TOA in a triangle without a right angle.
Practice Questions
Frequently Asked Questions
Do these ratios work in any triangle?
No. SOH CAH TOA only works in right-angled triangles. Other triangles need the sine rule or cosine rule.
How do I choose between sin, cos and tan?
Label the three sides, mark the two you are using (one known, one unknown), and pick the ratio that links them.
Keep Going With These Guides
Sine Rule and Cosine Rule: Formulas and Worked Examples
Solve non-right-angled triangles with the sine and cosine rules (law of sines and law of cosines) and find a triangle's area, with examples.
GCSEA-LevelHigh SchoolPythagoras' Theorem: How to Find a Missing Side
Use a² + b² = c² to find any side of a right-angled triangle. Worked examples for GCSE, and the Pythagorean theorem in US Geometry.
KS3GCSEMiddle SchoolHigh SchoolCircle Theorems Explained: Rules, Examples and Reasons
Learn the main circle theorems and use them to find angles with a reason for every step. GCSE Higher and high school Geometry, with worked examples.
GCSEA-LevelHigh SchoolSources and Review
- DfE: GCSE mathematics subject content and assessment objectives
- Common Core State Standards: Mathematics (high school)
Reviewed by StudyMorrow editorial review on 5 Nov 2026.