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GCSE Maths / Edexcel

Basic trigonometry

Label a right-angled triangle from the marked angle, choose the correct SOH CAH TOA ratio, and find missing sides or angles.

Geometry and MeasuresFoundation and HigherGrades 5 to 7Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Basic trigonometry

Pearson Edexcel GCSE Maths geometry G20: know and use trigonometric ratios in right-angled triangles.

Practise this topicUpload your answerAsk AI to explain thisTry exam-style questions
Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

Trigonometry in GCSE starts with right-angled triangles.

Label the side opposite the angle, the adjacent side next to the angle, and the hypotenuse opposite the right angle.

SOH means sin(theta) = opposite / hypotenuse.

CAH means cos(theta) = adjacent / hypotenuse.

TOA means tan(theta) = opposite / adjacent.

Choose the ratio that uses the side you know and the side you want. Do not pick sin, cos or tan because it feels familiar.

Opposite and adjacent change if the marked angle changes. Always label from the angle in the question.

If you are finding a side, write a trig equation and rearrange it.

If you are finding an angle, use inverse sin, inverse cos or inverse tan.

Make sure your calculator is in degree mode for GCSE angle questions.

Key ruleSOH CAH TOA: sin = O/H, cos = A/H, tan = O/A. For an angle, use inverse trig.

Diagram guide

θadjacentoppositehypotenuseSOH CAH TOA
Label sides from the angleOpposite and adjacent depend on the marked angle. The hypotenuse is always opposite the right angle.
θadjacentoppositehypotenuseSOHsin = O/HCAHcos = A/HTOAtan = O/A
Choose the ratioCircle the side you know and the side you want. Choose the ratio that contains both.
x8 cm6 cmfinding an angle needs inverse trigtan x = 6 ÷ 8x = tan⁻¹(6 ÷ 8)
Find an angleWhen the unknown is an angle, use inverse trigonometry after writing the correct ratio.

Worked examples

Find a side with sine

A right triangle has angle 30 degrees and hypotenuse 10 cm. Find the opposite side.

θadjacentoppositehypotenuseSOHsin = O/HCAHcos = A/HTOAtan = O/A
Example: sine uses opposite and hypotenuseThe known side is hypotenuse and the wanted side is opposite, so use sine.
  1. Use sin(theta) = opposite / hypotenuse.
  2. sin 30 = opposite / 10.
  3. Opposite = 10 sin 30.

Answer: 5 cm

Find a side with tangent

A right triangle has angle 40 degrees and adjacent side 8 cm. Find the opposite side.

θadjacentoppositehypotenuseSOH CAH TOA
Example: tangent uses opposite and adjacentTangent connects the opposite and adjacent sides.
  1. Use tan(theta) = opposite / adjacent.
  2. tan 40 = opposite / 8.
  3. Opposite = 8 tan 40.

Answer: 6.7 cm to 1 decimal place

Find an angle

Opposite = 6 cm and adjacent = 8 cm. Find the angle.

x8 cm6 cmfinding an angle needs inverse trigtan x = 6 ÷ 8x = tan⁻¹(6 ÷ 8)
Example: inverse tangentThe unknown is the angle, so use inverse tangent.
  1. Use tan(theta) = opposite / adjacent.
  2. tan(theta) = 6 / 8.
  3. theta = tan^-1(6 / 8).

Answer: 36.9 degrees to 1 decimal place

Choose the ratio before calculating

A right triangle has angle 55 degrees, opposite side 9 cm and unknown hypotenuse h. Which ratio should be used?

θadjacentoppositehypotenuseSOHsin = O/HCAHcos = A/HTOAtan = O/A
Example: choose firstOpposite and hypotenuse means sine.
  1. The sides involved are opposite and hypotenuse.
  2. SOH uses opposite and hypotenuse.
  3. Use sin 55 = 9 / h.

Answer: Use sine: sin 55 = 9 / h

Common mistakes

  • Labelling opposite and adjacent from the wrong angle.
  • Using the hypotenuse as adjacent.
  • Choosing the wrong trig ratio.
  • Forgetting inverse sin, cos or tan when finding an angle.
  • Using radians instead of degrees on the calculator.
  • Rounding too early before the final answer.

Quick exercise

Try these before moving to the exam-style questions.

  1. Which side is always opposite the right angle?
    θadjacentoppositehypotenuseSOH CAH TOA
    Quick check: hypotenuseThe hypotenuse is always opposite the right angle.
  2. Use sin 30 = x / 12. Find x.
    θadjacentoppositehypotenuseSOHsin = O/HCAHcos = A/HTOAtan = O/A
    Quick check: sine sideMultiply both sides by 12.
  3. Use tan 45 = x / 7. Find x.
    θadjacentoppositehypotenuseSOH CAH TOA
    Quick check: tangent sideTangent connects opposite and adjacent.
  4. Use cos 60 = x / 10. Find x.
    θadjacentoppositehypotenuseSOHsin = O/HCAHcos = A/HTOAtan = O/A
    Quick check: cosine sideCosine connects adjacent and hypotenuse.
  5. If opposite = 5 and hypotenuse = 13, which ratio finds the angle?
    x8 cm6 cmfinding an angle needs inverse trigtan x = 6 ÷ 8x = tan⁻¹(6 ÷ 8)
    Quick check: angle ratioOpposite and hypotenuse means sine, then inverse sine.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundation and HigherCalculator3 marks

In a right-angled triangle, an angle is 35 degrees and the hypotenuse is 12 cm. Find the side opposite the angle to 1 decimal place.

θadjacentoppositehypotenuseSOHsin = O/HCAHcos = A/HTOAtan = O/A
Question diagram: sine sideOpposite and hypotenuse means sine.
trigonometrysinemissing side
Standard exam styleFoundation and HigherCalculator4 marks

A right-angled triangle has angle 52 degrees and adjacent side 9 cm. Find the opposite side to 1 decimal place.

θadjacentoppositehypotenuseSOH CAH TOA
Question diagram: tangent sideOpposite and adjacent means tangent.
trigonometrytangentmethod marks
ChallengeHigherCalculator5 marks

In a right-angled triangle, the side opposite angle x is 7 cm and the hypotenuse is 15 cm. Find x to 1 decimal place.

x8 cm6 cmfinding an angle needs inverse trigtan x = 6 ÷ 8x = tan⁻¹(6 ÷ 8)
Question diagram: inverse sineThe unknown is an angle, so use inverse trig.
inverse trigonometrymissing anglehigher geometry