GCSE MATHS • HIGHER TIER

Sine Rule Made Simple

Find missing sides and angles in non-right-angled triangles using matching side-angle pairs.

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Sine Rule in Four Moves

1

Label

Label each side and its directly opposite angle carefully.

2

Match

Identify a known matching side-angle pair.

3

Substitute

Write the Sine Rule and substitute the known values.

4

Calculate

Rearrange, calculate and check that the answer is sensible.

What You’ll Learn

Understand what the Sine Rule is.

Recognise when to use the Sine Rule.

Identify matching opposite sides and angles.

Find missing sides in non-right-angled triangles.

Use degree mode correctly on a calculator.

Distinguish Sine Rule questions from other trigonometry methods.

Key Rules

Sine Rule Formula

a/sin(A) = b/sin(B) = c/sin(C)

Opposite pairs

Each side must be matched with angle directly opposite it.

When to use

Use the Sine Rule for a non-right-angled triangle when you know a matching side-angle pair.

Finding angles

Rearrange to find the sine value, then use sin⁻¹.

Worked Proof: Sine Rule

Step-by-Step Method

  • 1. Label the triangle.
  • 2. Identify matching side-angle pairs.
  • 3. Write the Sine Rule.
  • 4. Substitute known values.
  • 5. Rearrange carefully.
  • 6. Use a calculator.
  • 7. Check the answer is sensible.

More Worked Examples

1

Find side c when:

A = 50°, C = 80°, a = 10 cm.

10/sin50 = c/sin80
c ≈ 12.9 cm
Answer: 12.9 cm

2

Find angle B when:

a = 12 cm, b = 15 cm, A = 40°.

12/sin40 = 15/sinB
sinB = (15 × sin40) ÷ 12
≈ 0.804
B = sin⁻¹(0.804)
≈ 53.5°
Answer: 53.5°

3

Triangle angles are 50°, 60° and 70°. Side opposite 50° is 7 cm. Find side opposite 70°.

7/sin50 = x/sin70

x ≈ 8.6 cm
Answer: 8.6 cm

4

Side a = 6 cm, side b = 9 cm and angle A = 30°.

6/sin30 = 9/sinB

sinB = 0.75
B ≈ 48.6°
Answer: 48.6°
  • Common GCSE Mistakes
  • Pairing the wrong side and angle.
  • Using the Cosine Rule instead.
  • Forgetting the inverse sine button.
  • Calculator in radians.
  • Rounding too early.
  • Exam Tips
  • Identify a matching side-angle pair first.
  • Pair each side with the angle directly opposite it.
  • Use sin⁻¹ when finding an angle.
  • Keep the calculator in degree mode.
  • Avoid rounding too early.
  • Check that the result is sensible.

Skills Used in Sine Rule

Trigonometry

Non-Right-Angled Triangles

Opposite Side-Angle Pairs

Inverse Sine

Geometry

Calculator Skills

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Printable Worksheet

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Interactive Quiz

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Exam-Style Questions

Timed questions to build confidence for the real exam.

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Watch the Sine Rule Tutorial

Step-by-step walkthroughs of Sine Rule with clear methods and exam tips.

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Frequently Asked Questions

It is a formula linking sides and their opposite angles in a triangle.

Use it for a non-right-angled triangle when you have a matching side-angle pair.

Each lowercase side is paired with the uppercase angle directly opposite it.

Rearrange to find the sine value, then use sin⁻¹ on a calculator.

Degree mode, not radian mode.

Correctly identify the matching opposite side-angle pairs.