GCSE MATHS • HIGHER TIER
SOHCAHTOA Made Simple
Find missing sides and angles in right-angled triangles using sine, cosine and tangent.
Clear method
Exam practice
Worked proofs
SOHCAHTOA in Four Moves
1
Label
Identify the hypotenuse, opposite and adjacent sides.
2
Choose
Choose SOH, CAH or TOA using the sides you know and need.
3
Substitute
Write the trig equation and substitute the known values.
4
Calculate
Rearrange if needed, calculate and give a sensible rounded answer.
What You’ll Learn
Understand what SOHCAHTOA means.
Identify opposite, adjacent and hypotenuse correctly.
Choose the correct trigonometric ratio.
Find missing sides in right-angled triangles.
Find missing angles using inverse trig functions.
Use degree mode correctly on a calculator.
Apply SOHCAHTOA to GCSE and real-life problems.
Key Rules
SOH
sin = Opposite ÷ Hypotenuse
CAH
cos = Adjacent ÷ Hypotenuse
TOA
tan = Opposite ÷ Adjacent
Finding angles
Use sin⁻¹, cos⁻¹ or tan⁻¹ when the unknown is an angle. Keep the calculator in degree mode.
Worked Proof: SOHCAHTOA
Step-by-Step Method
- 1. Draw and label the triangle.
- 2. Identify opposite, adjacent and hypotenuse.
- 3. Choose SOH, CAH or TOA.
- 4. Write the equation.
- 5. Rearrange if needed.
- 6. Use the calculator carefully.
- 7. Give the answer with units and sensible rounding.
More Worked Examples
1
Finding a Side with Cosine
A triangle has an angle of 60°, a hypotenuse of 20 cm and an unknown adjacent side x.
Use CAH:
cos 60° = x / 20
x = 20 × cos 60°
cos 60° = 0.5
x = 10
cos 60° = x / 20
x = 20 × cos 60°
cos 60° = 0.5
x = 10
2
Finding a Side with Tangent
A triangle has an angle of 45°, an adjacent side of 8 cm and an unknown opposite side x.
Use TOA:
tan 45° = x / 8
x = 8 × tan 45°
tan 45° = 1
x = 8
tan 45° = x / 8
x = 8 × tan 45°
tan 45° = 1
x = 8
3
Finding an Angle with Sine
Opposite side = 6 cm, hypotenuse = 10 cm.
sin θ = 6 / 10 = 0.6
θ = sin⁻¹(0.6)
θ ≈ 36.9°
θ = sin⁻¹(0.6)
θ ≈ 36.9°
4
Finding an Angle with Cosine
Adjacent side = 12 cm, hypotenuse = 13 cm.
cos θ = 12 / 13 ≈ 0.9231
θ = cos⁻¹(0.9231)
θ ≈ 22.6°
θ = cos⁻¹(0.9231)
θ ≈ 22.6°
- Common GCSE Mistakes
- Choosing the wrong ratio.
- Misidentifying opposite and adjacent.
- Forgetting inverse trig when finding angles.
- Using degrees/radians incorrectly.
- Making calculator input errors.
- Exam Tips
- Find the hypotenuse first: it is always opposite the right angle.
- Label opposite and adjacent relative to the angle being used.
- Write SOH, CAH and TOA before choosing a ratio.
- Use inverse trig only when finding an angle.
- Make sure the calculator is in degree mode.
Skills Used in SOHCAHTOA
Trigonometry
Non-Right-Angled Triangles
Opposite Side-Angle Pairs
Inverse Sine
Geometry
Calculator Skills
Ready to Practise?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the SOHCAHTOA Tutorial
Step-by-step walkthroughs of SOHCAHTOA with clear methods and exam tips.
Frequently Asked Questions
SOH means sin = opposite/hypotenuse, CAH means cos = adjacent/hypotenuse, and TOA means tan = opposite/adjacent.
SOHCAHTOA is used with right-angled triangles.
It is the longest side and is directly opposite the right angle.
Choose the ratio containing the side you know and the side you need.
Use an inverse trig function when the unknown is an angle.
Use degree mode for GCSE angle questions unless the question states otherwise.