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GCSE Maths → Geometry → Trigonometry

GCSE Trigonometry: SOH CAH TOA Step-by-Step Guide

GCSE trigonometry is used to find missing sides and angles in right-angled triangles. The main idea is to use the relationship between an angle and the sides of the triangle.

Before learning trigonometry, it helps to be confident with Pythagoras theorem, because both topics involve right-angled triangles.

Video explanation

A short Worthing Maths Tutor video explanation for GCSE trigonometry SOH CAH TOA can be embedded here later to improve student engagement and time on page.

What is SOH CAH TOA?

SOH CAH TOA helps you remember the three trigonometry ratios:

SOH: sin θ = opposite ÷ hypotenuse
CAH: cos θ = adjacent ÷ hypotenuse
TOA: tan θ = opposite ÷ adjacent

How to label the sides

  • The hypotenuse is the longest side, opposite the right angle.
  • The opposite side is opposite the angle you are using.
  • The adjacent side is next to the angle, but is not the hypotenuse.
Exam tip: The opposite and adjacent sides depend on which angle you are using. Always label the triangle from the angle in the question.

Finding a missing side

Example 1: Find the opposite side using sin

A right-angled triangle has angle 30° and hypotenuse 10 cm. Find the opposite side.

sin θ = opposite ÷ hypotenuse
sin 30° = x ÷ 10
x = 10 × sin 30°
x = 5 cm

Example 2: Find the adjacent side using cos

A right-angled triangle has angle 60° and hypotenuse 12 cm. Find the adjacent side.

cos θ = adjacent ÷ hypotenuse
cos 60° = x ÷ 12
x = 12 × cos 60°
x = 6 cm

Example 3: Find the opposite side using tan

A right-angled triangle has angle 40° and adjacent side 8 cm. Find the opposite side.

tan θ = opposite ÷ adjacent
tan 40° = x ÷ 8
x = 8 × tan 40°
x = 6.7 cm to 1 decimal place

Finding a missing angle

To find a missing angle, use inverse sine, inverse cosine, or inverse tangent on your calculator.

Example 4: Find an angle using tan

A right-angled triangle has opposite side 5 cm and adjacent side 9 cm. Find the angle.

tan θ = opposite ÷ adjacent
tan θ = 5 ÷ 9
θ = tan⁻¹(5 ÷ 9)
θ = 29.1° to 1 decimal place

Example 5: Find an angle using sin

A right-angled triangle has opposite side 7 cm and hypotenuse 14 cm. Find the angle.

sin θ = opposite ÷ hypotenuse
sin θ = 7 ÷ 14
θ = sin⁻¹(0.5)
θ = 30°
Common mistake: Many students use sin, cos, or tan before labelling the sides. Always label opposite, adjacent, and hypotenuse first.

How to choose sin, cos, or tan

  1. Check that the triangle is right-angled.
  2. Mark the angle you are using.
  3. Label the hypotenuse, opposite, and adjacent sides.
  4. Look at the two sides involved in the question.
  5. Choose SOH, CAH, or TOA.

Common mistakes in GCSE trigonometry

  • Using trigonometry when the triangle is not right-angled.
  • Mixing up opposite and adjacent.
  • Forgetting to use inverse trig when finding an angle.
  • Using the wrong calculator mode.
  • Rounding too early before the final answer.
Exam tip: For GCSE questions, your calculator should normally be in degrees, not radians.

Practice questions

  1. A right-angled triangle has angle 30° and hypotenuse 8 cm. Find the opposite side.
  2. A right-angled triangle has angle 60° and hypotenuse 20 cm. Find the adjacent side.
  3. A right-angled triangle has angle 45° and adjacent side 10 cm. Find the opposite side.
  4. A right-angled triangle has opposite side 4 cm and adjacent side 7 cm. Find the angle.
  5. A right-angled triangle has opposite side 6 cm and hypotenuse 12 cm. Find the angle.

Answers

  1. 4 cm
  2. 10 cm
  3. 10 cm
  4. 29.7° to 1 decimal place
  5. 30°

GCSE trigonometry FAQ

What is SOH CAH TOA?

SOH CAH TOA helps you remember sine, cosine, and tangent ratios in right-angled triangles: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.

When do I use trigonometry?

Use trigonometry when you have a right-angled triangle and need to find a missing side or angle.

How do I know whether to use sin, cos, or tan?

Label the sides as opposite, adjacent, and hypotenuse relative to the angle. Then choose the ratio that uses the two sides involved.

Is trigonometry on GCSE maths?

Yes. Right-angled triangle trigonometry appears on both Foundation and Higher GCSE maths papers.

Related GCSE geometry topics

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