Graph of the cosine function.pptx

EDEXCEL GCSE MATHS · HIGHER

Graph of the cosine function

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Lesson 3 of 9

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EDEXCEL GCSE MATHS · HIGHER

Warm-up

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Lesson 3 of 9

Answer each one, then check.

1

What is cos 60°?

2

What is cos 0°?

3

What does CAH mean?

4

What is the period of the sine graph?

5

What are the largest and smallest values of sine?

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EDEXCEL GCSE MATHS · HIGHER

Warm-up - Answers

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Lesson 3 of 9

1

What is cos 60°?

0.5

2

What is cos 0°?

1

3

What does CAH mean?

cos = adjacent ÷ hypotenuse

4

What is the period of the sine graph?

360°

5

What are the largest and smallest values of sine?

1 and −1

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EDEXCEL GCSE MATHS · HIGHER

Learning Objectives

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Lesson 3 of 9

1

Sketch the graph of y = cos x for 0° ≤ x ≤ 360°.

2

Recall key values of cosine.

3

Use symmetry to find a second solution of cos x = k.

4

Describe how the cosine and sine graphs are related.

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EDEXCEL GCSE MATHS · HIGHER

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Lesson 3 of 9

COSINE GRAPH

The graph of y = cos x is the sine wave shifted 90° to the left.

It starts at 1 when x = 0°, falls to 0 at 90°, reaches −1 at 180°, is 0 at 270° and back to 1 at 360°.

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EDEXCEL GCSE MATHS · HIGHER

The Cosine Graph

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Lesson 3 of 9

Cosine x equals 0.5 has two solutions: 60 degrees and 300 degrees.

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EDEXCEL GCSE MATHS · HIGHER

Key Values of Cosine

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Lesson 3 of 9

These are worth learning.

x

0°

60°

90°

180°

cos x

1

0.5

0

−1 | 0 | 0.5 | 1

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EDEXCEL GCSE MATHS · HIGHER

Features of the Graph

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Lesson 3 of 9

Period

Repeats every 360°.

Maximum

1 at x = 0° and 360°.

Minimum

−1 at x = 180°.

Roots

x = 90° and 270°.

Symmetry

Symmetrical about x = 180°.

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EDEXCEL GCSE MATHS · HIGHER

Finding Two Solutions

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Lesson 3 of 9

Use symmetry about x = 180°.

1

Use the calculator

Find x = cos ⁻¹(k); this is the first solution

2

Reflect about 180

Second solution = 360°− x

3

Check the range

Both must be between 0° and 360°

4

Negative values

cos ⁻¹ of a negative gives an angle between 90° and 180°

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EDEXCEL GCSE MATHS · HIGHER

Solving cos x = 0.5

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Lesson 3 of 9

Solve cos x = 0.5 for 0° ≤ x ≤ 360°.

1

Calculator

x = cos ⁻¹(0.5) = 60°

2

Reflect

x = 360°− 60°= 300°

ANSWER

x = 60° and x = 300°

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Solving a Negative Value

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Lesson 3 of 9

Solve cos x = −0.7 for 0° ≤ x ≤ 360°, correct to 1 decimal place.

1

Calculator

x = cos ⁻¹(−0.7) = 134.4°

2

Reflect

360°− 134.4°= 225.6°

ANSWER

x = 134.4° and x = 225.6°

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EDEXCEL GCSE MATHS · HIGHER

Sine and Cosine Graphs

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Lesson 3 of 9

Y = SIN X

• Starts at 0.

• Maximum at 90°, minimum at 270°.

• Second solution: 180°− x.

Y = COS X

• Starts at 1.

• Maximum at 0° and 360°, minimum at 180°.

• Second solution: 360°− x.

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Key Terms

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Lesson 3 of 9

Cosine graph

The wave y = cos x.

Period

360° for both sine and cosine.

Inverse cosine

cos ⁻¹, the calculator function that finds an angle.

Translation

Sliding a graph without turning it.

Symmetry

The graph mirrors itself about a line.

Solution

A value of x that gives the required y.

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EDEXCEL GCSE MATHS · HIGHER

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Lesson 3 of 9

YOUR TASK

Sine Meets Cosine

12 minutes

Sketch y = sin x and y = cos x on the same axes for 0° to 360°. (a) Where do they cross? (b) What single translation maps sine onto cosine?

1

Draw both curves.

2

Read the crossing points.

WHAT A GOOD ANSWER SHOWS

(a) At x = 45° and 225°. (b) A translation of 90° to the left.

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Can I...?

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Lesson 3 of 9

Sketch the cosine graph.

State its maximum and minimum.

Recall cos 0, cos 60 and cos 90.

Use cos inverse.

Find the second solution.

Solve for negative values.

Compare sine and cosine.

Read solutions from a graph.

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EDEXCEL GCSE MATHS · HIGHER

Summary & Exam Focus

Cosine starts at 1, falls to −1 at 180°, returns to 1 at 360°.

cos x = k has two solutions in 0° to 360° for −1 < k < 1.

Solutions are x and 360°− x.

Cosine is sine shifted 90° left.

EXAM FOCUS

Solve cos x = 0.3 for 0° ≤ x ≤ 360°. Give answers correct to 1 decimal place. (3 marks)

Second solution is 360°− x.

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