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Maths · More trigonometry

The cosine rule and 2D trigonometric problems

Use the cosine rule to find a side or an angle when the sine rule cannot be used, and choose the right method in 2D problems including bearings.

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Teacher resources

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Student handouts

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Warm-up

Answer each one, then check.

  1. 1

    What is \(\cos 60^\circ\)?

    Show answerHide answer

    \(0.5\)

  2. 2

    What is Pythagoras' theorem?

    Show answerHide answer

    \(a^2 + b^2 = c^2\)

  3. 3

    What is the sine rule?

    Show answerHide answer

    \(\frac{a}{\sin A} = \frac{b}{\sin B}\)

  4. 4

    What is a bearing?

    Show answerHide answer

    An angle measured clockwise from north, with three figures

  5. 5

    What is \(\cos^{-1}(0.5)\)?

    Show answerHide answer

    \(60^\circ\)

Learning Objectives

  1. 1Use the cosine rule to find a missing side.
  2. 2Rearrange the cosine rule to find a missing angle.
  3. 3Decide between the sine rule, cosine rule and right-angled trigonometry.
  4. 4Solve 2D problems, including bearings.

COSINE RULE

The cosine rule links three sides and one angle: it is Pythagoras with a correction.

\(a^2 = b^2 + c^2 - 2bc\cos A\). To find an angle: \(\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}\).

Which Rule?

Use the SINE rule when...

  • You have two angles and a side.
  • You have two sides and an angle not between them.
  • You have a matching pair (a side and its opposite angle).

Use the COSINE rule when...

  • You have two sides and the angle between them.
  • You have all three sides.
  • You do not have a matching pair.

Cosine Rule: Finding a Side

In triangle ABC, \(b = 7\) cm, \(c = 9\) cm and angle \(A = 60^\circ\). Find \(a\).

Show the solutionHide the solution
  1. 1 Write the rule \(a^2 = b^2 + c^2 - 2bc\cos A\)
  2. 2 Substitute \(a^2 = 7^2 + 9^2 - 2 \times 7 \times 9 \times \cos 60^\circ\)
  3. 3 Work out \(a^2 = 49 + 81 - 63 = 67\)
  4. 4 Square root \(a = 8.19\)

Answer\(a = 8.19\) cm (3 s.f.)

Cosine Rule: Finding an Angle

A triangle has sides 5 cm, 7 cm and 9 cm. Find the largest angle.

Show the solutionHide the solution
  1. 1 Largest angle is opposite 9 \(a = 9\), \(b = 5\), \(c = 7\)
  2. 2 Rearranged rule \(\cos A = \dfrac{5^2 + 7^2 - 9^2}{2 \times 5 \times 7}\)
  3. 3 Work out \(\cos A = \dfrac{-7}{70} = -0.1\)
  4. 4 Inverse cosine \(A = 95.7^\circ\)

AnswerLargest angle \(= 95.7^\circ\) (1 d.p.)

A Bearings Problem

A ship sails 12 km on a bearing of \(040^\circ\) and then 9 km on a bearing of \(130^\circ\). How far is it from its starting point?

Show the solutionHide the solution
  1. 1 Angle between the two legs \(130^\circ - 40^\circ = 90^\circ\)
  2. 2 Right-angled triangle Use Pythagoras
  3. 3 Work out \(12^2 + 9^2 = 144 + 81 = 225\)
  4. 4 Square root \(\sqrt{225} = 15\)

AnswerThe ship is 15 km from its start.

Common Slips

Avoid these.

  • Subtracting before multiplying

    Work out \(2bc\cos A\) as a whole first, then subtract.

  • Wrong angle

    The angle in the formula is opposite the side on the left-hand side.

  • Rounding early

    Keep the full value until the final step.

  • Angle over 90

    A negative cosine means the angle is obtuse; this is fine.

Which Rule and Why?

For each triangle, state which rule you would use first. (a) \(a = 6\), \(b = 8\), \(C = 50^\circ\), find \(c\). (b) \(a = 5\), \(b = 7\), \(c = 9\), find \(A\). (c) \(A = 40^\circ\), \(B = 70^\circ\), \(a = 8\), find \(b\). (d) \(a = 9\), \(b = 12\), \(A = 30^\circ\), find \(B\).

1. Check for a side with its opposite angle.

2. If there is none, use the cosine rule.

A good answer shows: (a) Cosine rule (two sides and the included angle). (b) Cosine rule (three sides). (c) Sine rule. (d) Sine rule.

Can I...?

  1. 1Write the cosine rule.
  2. 2Find a missing side.
  3. 3Rearrange to find an angle.
  4. 4Identify the largest angle.
  5. 5Choose sine or cosine rule.
  6. 6Solve a bearings problem.
  7. 7Avoid rounding too early.
  8. 8Check with a sketch.

Summary & Exam Focus

  • \(a^2 = b^2 + c^2 - 2bc\cos A\).
  • \(\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}\).
  • Use cosine rule for SAS and SSS, sine rule when you have a matching pair.
  • Bearings: measure clockwise from north.

Exam focus

In triangle PQR, \(PQ = 8.4\) cm, \(PR = 6.1\) cm and angle \(QPR = 57^\circ\). Work out the length of \(QR\). Give your answer correct to 3 significant figures. (3 marks) (3 marks)

Write the rule first, substitute, then use your calculator in one go.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Cosine rule
\(a^2 = b^2 + c^2 - 2bc\cos A\).
Bearing
A direction measured clockwise from north as three figures.
Included angle
The angle between two given sides.
Obtuse
An angle between \(90^\circ\) and \(180^\circ\).
Largest angle
The angle opposite the longest side.
Rearrange
Change the subject of a formula.

Questions and answers

12 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Work out 3 marks Easier

In triangle PQR, \(PQ = 8.4\) cm, \(PR = 6.1\) cm and angle \(QPR = 57^\circ\). Work out the length of \(QR\). Give your answer correct to 3 significant figures.

Triangle PQR with PQ equal to 8.4 cm, PR equal to 6.1 cm and angle P equal to 57 degrees.

Mark scheme — 3 marks available

  • \(8.4^2 + 6.1^2 - 2 \times 8.4 \times 6.1 \times \cos 57^\circ\) — M1
  • 51.9... — A1
  • 7.21 — A1

Model answer

\(QR^2 = 8.4^2 + 6.1^2 - 2 \times 8.4 \times 6.1 \times \cos 57^\circ = 51.95\), so \(QR = 7.21\) cm

2. Exam question Work out 3 marks Easier

In triangle ABC, \(AB = 8\) cm, \(AC = 6\) cm and angle \(BAC = 40^\circ\). Work out the length of \(BC\). Give your answer correct to 3 significant figures.

Mark scheme — 3 marks available

  • \(6^2 + 8^2 - 2 \times 6 \times 8 \times \cos 40^\circ\) — M1
  • 26.4... — A1
  • 5.14 — A1

Model answer

\(BC^2 = 6^2 + 8^2 - 2 \times 6 \times 8 \times \cos 40^\circ = 26.46\), so \(BC = 5.14\) cm

3. Exam question Work out 3 marks Easier

A triangle has sides of length 6 cm, 8 cm and 11 cm. Work out the size of the largest angle. Give your answer correct to 1 decimal place.

Mark scheme — 3 marks available

  • \(\dfrac{6^2 + 8^2 - 11^2}{2 \times 6 \times 8}\) — M1
  • \(-0.21875\) — A1
  • 102.6 — A1

Model answer

\(\cos A = \dfrac{6^2 + 8^2 - 11^2}{2 \times 6 \times 8} = -0.21875\), so \(A = 102.6^\circ\)

4. Exam question Work out 3 marks Easier

A ship sails 12 km on a bearing of \(040^\circ\). It then sails 9 km on a bearing of \(130^\circ\). Work out the distance of the ship from its starting point.

Mark scheme — 3 marks available

  • Right angle identified — M1
  • \(12^2 + 9^2 = 225\) — M1
  • 15 — A1

Model answer

The angle between the two legs is \(130^\circ - 40^\circ = 90^\circ\), so the distance is \(\sqrt{12^2 + 9^2} = \sqrt{225} = 15\) km.

5. Exam question Work out 3 marks Easier

A triangle has sides of length 7 cm, 8 cm and 9 cm. Work out the size of the smallest angle. Give your answer correct to 1 decimal place.

Mark scheme — 3 marks available

  • \(\dfrac{8^2 + 9^2 - 7^2}{2 \times 8 \times 9}\) — M1
  • 0.666... — A1
  • 48.2 — A1

Model answer

\(\cos A = \dfrac{8^2 + 9^2 - 7^2}{2 \times 8 \times 9} = 0.6667\), so \(A = 48.2^\circ\)

6. Exam question Explain 2 marks Easier

In triangle ABC, you are given \(a\), \(c\) and angle \(B\). Which rule would you use to find \(b\), the sine rule or the cosine rule? Give a reason.

Mark scheme — 2 marks available

  • Cosine rule — B1
  • Two sides and the included angle — B1

Model answer

The cosine rule, because two sides and the angle between them are known and there is no matching side and angle pair.

7. Multiple choice 1 mark Easier

The cosine rule is...

  1. A \(a^2 = b^2 + c^2 + 2bc\cos A\)
  2. B \(a^2 = b^2 + c^2 - 2bc\cos A\) Correct
  3. C \(a = b + c - 2bc\cos A\)
  4. D \(a^2 = b^2 - c^2 - 2bc\cos A\)

Why: \(a^2 = b^2 + c^2 - 2bc\cos A\).

8. Multiple choice 1 mark Core

When two sides and the angle between them are known, use the...

  1. A Sine rule
  2. B Area formula only
  3. C Cosine rule Correct
  4. D Pythagoras

Why: Cosine rule finds the third side.

9. Multiple choice 1 mark Core

A triangle has sides 5, 7 and 9. To find the largest angle, use...

  1. A The cosine rule Correct
  2. B The sine rule
  3. C SOH CAH TOA
  4. D The area formula

Why: Three sides given: cosine rule, with the angle opposite 9.

10. Multiple choice 1 mark Core

If \(\cos A\) is negative, angle \(A\) is...

  1. A Acute
  2. B A right angle
  3. C Zero
  4. D Obtuse Correct

Why: Cosine is negative for angles between 90 and 180 degrees.

11. Multiple choice 1 mark Core

Bearings are measured...

  1. A Anticlockwise from north
  2. B Clockwise from north Correct
  3. C Clockwise from east
  4. D From south

Why: Clockwise from north, with three figures.

12. Multiple choice 1 mark Stretch

When \(A = 90^\circ\), the cosine rule becomes...

  1. A \(a = b + c\)
  2. B \(a^2 = b^2 - c^2\)
  3. C \(a^2 = b^2 + c^2\) Correct
  4. D \(a^2 = 2bc\)

Why: \(\cos 90^\circ = 0\), so \(a^2 = b^2 + c^2\): Pythagoras.