The cosine rule and 2D trigonometric problems - Teacher Slides.pptx

EDEXCEL GCSE MATHS · HIGHER

The cosine rule and 2D trigonometric problems

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Lesson 6 of 9

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Warm-up

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Lesson 6 of 9

Answer each one, then check.

1

What is cos 60°?

2

What is Pythagoras' theorem?

3

What is the sine rule?

4

What is a bearing?

5

What is cos ⁻¹(0.5)?

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Warm-up - Answers

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Lesson 6 of 9

1

What is cos 60°?

0.5

2

What is Pythagoras' theorem?

a² + b² = c²

3

What is the sine rule?

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

4

What is a bearing?

An angle measured clockwise from north, with three figures

5

What is cos ⁻¹(0.5)?

60°

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Learning Objectives

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Lesson 6 of 9

1

Use the cosine rule to find a missing side.

2

Rearrange the cosine rule to find a missing angle.

3

Decide between the sine rule, cosine rule and right-angled trigonometry.

4

Solve 2D problems, including bearings.

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Lesson 6 of 9

COSINE RULE

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

a² = b² + c² − 2bccos A. To find an angle: cos A = (b² + c² − a²)/2bc.

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The Cosine Rule

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Lesson 6 of 9

The angle in the formula is opposite the side you are finding.

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Which Rule?

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Lesson 6 of 9

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.

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Cosine Rule: Finding a Side

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Lesson 6 of 9

In triangle ABC, b = 7 cm, c = 9 cm and angle A = 60°. Find a.

1

Write the rule

a² = b² + c² − 2bccos A

2

Substitute

a² = 7² + 9² − 2 × 7 × 9 × cos 60°

3

Work out

a² = 49 + 81 − 63 = 67

4

Square root

a = 8.19

ANSWER

a = 8.19 cm (3 s.f.)

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Cosine Rule: Finding an Angle

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Lesson 6 of 9

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

1

Largest angle is opposite 9

a = 9, b = 5, c = 7

2

Rearranged rule

cos A = (5² + 7² − 9²)/(2 × 5 × 7)

3

Work out

cos A = (−7)/70 = −0.1

4

Inverse cosine

A = 95.7°

ANSWER

Largest angle = 95.7° (1 d.p.)

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A Bearings Problem

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Lesson 6 of 9

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

1

Angle between the two legs

130°− 40°= 90°

2

Right-angled triangle

Use Pythagoras

3

Work out

12² + 9² = 144 + 81 = 225

4

Square root

√225 = 15

ANSWER

The ship is 15 km from its start.

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Common Slips

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Lesson 6 of 9

Avoid these.

Subtracting before multiplying

Work out 2bccos 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.

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Key Terms

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Lesson 6 of 9

Cosine rule

a² = b² + c² − 2bccos A.

Bearing

A direction measured clockwise from north as three figures.

Included angle

The angle between two given sides.

Obtuse

An angle between 90° and 180°.

Largest angle

The angle opposite the longest side.

Rearrange

Change the subject of a formula.

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Lesson 6 of 9

YOUR TASK

Which Rule and Why?

12 minutes

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

1

Check for a side with its opposite angle.

2

If there is none, use the cosine rule.

WHAT A GOOD ANSWER SHOWS

(a) Cosine rule (two sides and the included angle). (b) Cosine rule (three sides). (c) Sine rule. (d) Sine rule.

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Can I...?

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Lesson 6 of 9

Write the cosine rule.

Find a missing side.

Rearrange to find an angle.

Identify the largest angle.

Choose sine or cosine rule.

Solve a bearings problem.

Avoid rounding too early.

Check with a sketch.

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Summary & Exam Focus

a² = b² + c² − 2bccos A.

cos A = (b² + c² − a²)/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°. Work out the length of QR. Give your answer correct to 3 significant figures. (3 marks)

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

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Exam Practice: The cosine rule

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Lesson 6 of 9

Answer all questions. Show your working. · 30 minutes

Question 1 · 3 marks · Work out

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

Question 2 · 3 marks · Work out

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

Question 3 · 3 marks · Work out

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.

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Exam Practice: The cosine rule (cont.)

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Lesson 6 of 9

Question 4 · 3 marks · Work out

A ship sails 12 km on a bearing of 040°. It then sails 9 km on a bearing of 130°. Work out the distance of the ship from its starting point.

Question 5 · 3 marks · Work out

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.

Question 6 · 2 marks · Explain

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.

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Question 1 · 3 marks · Work out

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Lesson 6 of 9

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

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Question 1 · mark scheme

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Lesson 6 of 9

3 marks available. Award a mark for each point made.

8.4² + 6.1² − 2 × 8.4 × 6.1 × cos 57°

M1

51.9...

A1

7.21

A1

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Question 2 · 3 marks · Work out

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Lesson 6 of 9

“

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

HOW TO ANSWER IT

Command word: Work out. Worth 3 marks, so plan before writing.

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Question 2 · mark scheme

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Lesson 6 of 9

3 marks available. Award a mark for each point made.

6² + 8² − 2 × 6 × 8 × cos 40°

M1

26.4...

A1

5.14

A1

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Question 3 · 3 marks · Work out

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Lesson 6 of 9

“

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.

HOW TO ANSWER IT

Command word: Work out. Worth 3 marks, so plan before writing.

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Question 3 · mark scheme

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Lesson 6 of 9

3 marks available. Award a mark for each point made.

(6² + 8² − 11²)/(2 × 6 × 8)

M1

−0.21875

A1

102.6

A1

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Question 4 · 3 marks · Work out

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Lesson 6 of 9

“

A ship sails 12 km on a bearing of 040°. It then sails 9 km on a bearing of 130°. Work out the distance of the ship from its starting point.

HOW TO ANSWER IT

Command word: Work out. Worth 3 marks, so plan before writing.

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Question 4 · mark scheme

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Lesson 6 of 9

3 marks available. Award a mark for each point made.

Right angle identified

M1

12² + 9² = 225

M1

15

A1

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Question 5 · 3 marks · Work out

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Lesson 6 of 9

“

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.

HOW TO ANSWER IT

Command word: Work out. Worth 3 marks, so plan before writing.

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Question 5 · mark scheme

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Lesson 6 of 9

3 marks available. Award a mark for each point made.

(8² + 9² − 7²)/(2 × 8 × 9)

M1

0.666...

A1

48.2

A1

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Question 6 · 2 marks · Explain

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Lesson 6 of 9

“

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.

HOW TO ANSWER IT

Command word: Explain. Worth 2 marks, so plan before writing.

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Question 6 · mark scheme

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Lesson 6 of 9

2 marks available. Award a mark for each point made.

Cosine rule

B1

Two sides and the included angle

B1

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