GCSE · GCSE Maths
GCSE Maths trigonometry: SOHCAHTOA
What is sohcahtoa?
SOHCAHTOA is the mnemonic for right-angled triangle trig: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. GCSE Higher students use it to find unknown sides or angles in any right-angled triangle. For non-right-angled triangles, switch to the sine rule or cosine rule.
Worked example
A right-angled triangle has hypotenuse 10 cm and the angle opposite the unknown side is 30°. Find the unknown side.
- Identify the angle (30°), the hypotenuse (10 cm) and the unknown side (opposite the 30° angle).
- Pick the right ratio: opposite and hypotenuse → use sine. sin(30°) = opposite ÷ 10.
- Rearrange: opposite = 10 × sin(30°).
- Evaluate: sin(30°) = 0.5, so opposite = 10 × 0.5 = 5 cm.
When to use SOHCAHTOA
Use it in any right-angled triangle when you know one angle (not the right angle) plus one side, and you want to find another side; or when you know two sides and want to find an angle.
Finding an angle
Pick the ratio that uses the two sides you know. Apply the inverse trig function (sin⁻¹, cos⁻¹, tan⁻¹) on a calculator. Most calculators need DEG mode set — RAD mode gives wildly wrong answers.
Where it appears in the GCSE exam
AQA 8300, Edexcel 1MA1 and OCR J560 all examine SOHCAHTOA on Higher tier (and the easier cases on Foundation tier). It frequently combines with Pythagoras' theorem in multi-step questions.
Common mistakes
- Forgetting to set the calculator to degrees (DEG) — radians give wrong answers.
- Mixing up opposite and adjacent. The opposite side is across from the named angle; the adjacent is next to it.
- Using SOHCAHTOA on a non-right-angled triangle. Switch to the sine or cosine rule instead.
- Rounding too early — keep extra decimal places until the final answer.
Frequently asked
- What's the difference between sin, cos and tan?
- All three are ratios of side lengths in a right-angled triangle. Sin pairs opposite with hypotenuse; cos pairs adjacent with hypotenuse; tan pairs opposite with adjacent. Use the SOHCAHTOA mnemonic to pick the right one.
- When do I use the sine or cosine rule instead?
- Use sine/cosine rule for non-right-angled triangles. SOHCAHTOA only works when the triangle has a 90° angle.
- Do I need to memorise sin(30°), cos(60°), etc.?
- Higher tier students should know the exact values for 0°, 30°, 45°, 60° and 90°. These often appear in 'find the exact value' questions where a calculator answer (0.5) would be marked down.
GCSE Maths glossary terms
- SOHCAHTOASOHCAHTOA is the mnemonic for the three right-angled trig ratios: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. Use it in any right-angled triangle to find an unknown side or angle when you know one side + one angle (other than the right angle). For non-right-angled triangles, use the sine rule or cosine rule instead.
- Pythagoras' theoremPythagoras' theorem states that in a right-angled triangle, the square on the hypotenuse equals the sum of squares on the other two sides: a² + b² = c². It applies only to right-angled triangles. GCSE questions extend it to 3D problems, isosceles triangles split into two right-angled halves, and finding distances between coordinates.
- Sine ruleThe sine rule relates the sides and opposite angles of any triangle: a / sin A = b / sin B = c / sin C. Use it when you know two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA, the ambiguous case). For right-angled triangles, SOHCAHTOA is simpler and preferred.
- Cosine ruleThe cosine rule relates the three sides and one angle of any triangle: a² = b² + c² − 2bc · cos A. Use it when you know two sides and the included angle (SAS), or all three sides and want to find an angle (SSS). For right-angled triangles, Pythagoras' theorem (cos 90° = 0) is the natural special case.
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