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.
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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).
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