Graph of the sine function - Teacher Slides.pptx

EDEXCEL GCSE MATHS · HIGHER

Graph of the sine function

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Lesson 2 of 9

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EDEXCEL GCSE MATHS · HIGHER

Warm-up

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Lesson 2 of 9

Answer each one, then check.

1

What is sin 30°?

2

What is sin 90°?

3

What is the largest value sin x can take?

4

What does SOH mean?

5

Which mode must a calculator be in for degrees?

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EDEXCEL GCSE MATHS · HIGHER

Warm-up - Answers

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Lesson 2 of 9

1

What is sin 30°?

0.5

2

What is sin 90°?

1

3

What is the largest value sin x can take?

1

4

What does SOH mean?

sin = opposite ÷ hypotenuse

5

Which mode must a calculator be in for degrees?

DEG

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EDEXCEL GCSE MATHS · HIGHER

Learning Objectives

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Lesson 2 of 9

1

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

2

Recall key values of sine.

3

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

4

Read solutions from the graph.

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EDEXCEL GCSE MATHS · HIGHER

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Lesson 2 of 9

SINE GRAPH

The graph of y = sin x is a smooth wave that repeats every 360°.

It starts at 0, rises to 1 at 90°, returns to 0 at 180°, falls to −1 at 270° and returns to 0 at 360°.

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EDEXCEL GCSE MATHS · HIGHER

The Sine Graph

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Lesson 2 of 9

Sine x equals 0.5 has two solutions: 30 degrees and 150 degrees.

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EDEXCEL GCSE MATHS · HIGHER

Key Values of Sine

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Lesson 2 of 9

These are worth learning.

x

0°

30°

90°

150°

sin x

0

0.5

1

0.5 | 0 | −1 | 0

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Features of the Graph

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Lesson 2 of 9

Period

Repeats every 360°.

Maximum

1 at x = 90°.

Minimum

−1 at x = 270°.

Roots

x = 0°, 180°, 360°.

Symmetry

Symmetrical about x = 90° and x = 270°.

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EDEXCEL GCSE MATHS · HIGHER

Finding Two Solutions

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Lesson 2 of 9

Use symmetry about x = 90°.

1

Use the calculator

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

2

Reflect about 90

Second solution = 180°− x

3

Check the range

Both must be between 0° and 360°

4

Negative values

If k < 0, the solutions are between 180° and 360°

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EDEXCEL GCSE MATHS · HIGHER

Solving sin x = 0.5

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Lesson 2 of 9

Solve sin x = 0.5 for 0° ≤ x ≤ 360°.

1

Calculator

x = sin ⁻¹(0.5) = 30°

2

Reflect

x = 180°− 30°= 150°

ANSWER

x = 30° and x = 150°

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Solving a Negative Value

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Lesson 2 of 9

Solve sin x = −0.5 for 0° ≤ x ≤ 360°.

1

Calculator gives

sin ⁻¹(−0.5) = −30°, so use the positive 30° as a reference

2

Below the axis

Solutions are 180°+ 30° and 360°− 30°

3

Values

210° and 330°

ANSWER

x = 210° and x = 330°

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Key Terms

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Lesson 2 of 9

Period

The length after which the graph repeats.

Amplitude

The height from the middle line to the maximum.

Maximum

The highest value on the graph.

Minimum

The lowest value on the graph.

Inverse sine

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

Symmetry

The graph looks the same on both sides of a line.

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EDEXCEL GCSE MATHS · HIGHER

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Lesson 2 of 9

YOUR TASK

Sketch and Solve

12 minutes

(a) Sketch y = sin x for 0° to 360°. (b) Mark y = 0.8 and read the two solutions from your sketch. (c) Calculate them: sin ⁻¹(0.8) = 53.1°.

1

Draw the axes with 90° marks.

2

Use symmetry about 90°.

WHAT A GOOD ANSWER SHOWS

(b) About 53° and 127°. (c) x = 53.1° and 180°− 53.1°= 126.9°.

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Can I...?

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Lesson 2 of 9

Sketch the sine graph.

State its maximum and minimum.

State its period.

Recall sin 30 and sin 90.

Use sin inverse.

Find the second solution.

Solve for negative values.

Read solutions from a graph.

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Summary & Exam Focus

Sine repeats every 360°, max 1, min −1.

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

Positive k: x and 180°− x.

Negative k: angles in the second half.

EXAM FOCUS

Solve sin x = 0.6 for 0° ≤ x ≤ 360°. Give answers correct to 1 decimal place. (3 marks)

Second solution is 180°− x. Check both are in range.

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Exam Practice: Graph of the sine function

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Lesson 2 of 9

Answer all questions. Show your working. · 25 minutes

Question 1 · 3 marks · Solve

Solve sin x = 0.6 for 0° ≤ x ≤ 360°. Give your answers correct to 1 decimal place.

Question 2 · 2 marks · Write down

Write down the value of (a) sin 90° (b) sin 270°

Question 3 · 3 marks · Use the graph

The graph shows y = sin x for 0° ≤ x ≤ 360°. Use the graph to solve sin x = 0.5 and to write down the maximum value of sin x.

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EDEXCEL GCSE MATHS · HIGHER

Exam Practice: Graph of the sine function (cont.)

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Lesson 2 of 9

Question 4 · 3 marks · Solve

Solve sin x = −0.3 for 0° ≤ x ≤ 360°. Give your answers correct to 1 decimal place.

Question 5 · 2 marks · Explain

Amir says that sin x = 1.4 has a solution. Explain why he is wrong.

Question 6 · 2 marks · Write down

sin 40°= 0.64 (to 2 decimal places). Write down another angle between 0° and 360° with sine 0.64.

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Question 1 · 3 marks · Solve

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Lesson 2 of 9

“

Solve sin x = 0.6 for 0° ≤ x ≤ 360°. Give your answers correct to 1 decimal place.

HOW TO ANSWER IT

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

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Question 1 · mark scheme

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Lesson 2 of 9

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

sin ⁻¹(0.6) = 36.9°

B1

180 − 36.9

M1

143.1

A1

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Question 2 · 2 marks · Write down

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Lesson 2 of 9

“

Write down the value of (a) sin 90° (b) sin 270°

HOW TO ANSWER IT

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

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Question 2 · mark scheme

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Lesson 2 of 9

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

1

B1

−1

B1

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Question 3 · 3 marks · Use the graph

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Lesson 2 of 9

The graph shows y = sin x for 0° ≤ x ≤ 360°. Use the graph to solve sin x = 0.5 and to write down the maximum value of sin x. (3 marks)

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Question 3 · mark scheme

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Lesson 2 of 9

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

30

B1

150

B1

1

B1

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Question 4 · 3 marks · Solve

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Lesson 2 of 9

“

Solve sin x = −0.3 for 0° ≤ x ≤ 360°. Give your answers correct to 1 decimal place.

HOW TO ANSWER IT

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

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Question 4 · mark scheme

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Lesson 2 of 9

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

Reference angle 17.5°

B1

180 + 17.5 or 360 − 17.5

M1

Both answers

A1

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Question 5 · 2 marks · Explain

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Lesson 2 of 9

“

Amir says that sin x = 1.4 has a solution. Explain why he is wrong.

HOW TO ANSWER IT

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

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Question 5 · mark scheme

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Lesson 2 of 9

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

Maximum of sine is 1

M1

Conclusion

C1

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EDEXCEL GCSE MATHS · HIGHER

Question 6 · 2 marks · Write down

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Lesson 2 of 9

“

sin 40°= 0.64 (to 2 decimal places). Write down another angle between 0° and 360° with sine 0.64.

HOW TO ANSWER IT

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

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Question 6 · mark scheme

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Lesson 2 of 9

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

180 − 40

M1

140

A1

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