EDEXCEL GCSE MATHS · HIGHER
Graph of the sine function
More trigonometry · Lesson 2 of 9
Teacher copy - includes the notes for whoever is teaching from it.
Warm-up
Answer each one, then check.
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
Learning Objectives
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.
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°.
The Sine Graph
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Sine x equals 0.5 has two solutions: 30 degrees and 150 degrees. |
The graph of y equals sine x from 0 to 360 degrees, showing the maximum at 90, the minimum at 270 and the two solutions of sine x equals 0.5.
Key Values of Sine
These are worth learning.
|
x |
0° |
30° |
90° |
150° |
|---|---|---|---|---|
|
sin x |
0 |
0.5 |
1 |
0.5 | 0 | −1 | 0 |
Features of the Graph
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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°. |
|
Finding Two Solutions
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° |
Solving sin x = 0.5
|
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°
Solving a Negative Value
|
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°
Key Terms
|
Period The length after which the graph repeats. |
Amplitude The height from the middle line to the maximum. |
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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. |
Your Task: Sketch and Solve
12 minutes
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(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°. |
A good answer shows: (b) About 53° and 127°. (c) x = 53.1° and 180°− 53.1°= 126.9°.
Note: Check that the sketch solutions match the calculated ones.
Can I...?
☐ 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.
Summary
✓ 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.
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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. |
Exam Practice: Graph of the sine function
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.
▸ 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.
Question 1 · 3 marks · Solve
|
“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.
Question 1 · mark scheme
3 marks available. Award a mark for each point made.
▸ sin ⁻¹(0.6) = 36.9°. B1
▸ 180 − 36.9. M1
▸ 143.1. A1
▸ Model answer. x = 36.9° and x = 143.1°
Question 2 · 2 marks · Write down
|
“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.
Question 2 · mark scheme
2 marks available. Award a mark for each point made.
▸ 1. B1
▸ −1. B1
▸ Model answer. (a) 1 (b) −1
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. (3 marks) |
|
Question 3 · mark scheme
3 marks available. Award a mark for each point made.
▸ 30. B1
▸ 150. B1
▸ 1. B1
▸ Model answer. x = 30° and x = 150°; maximum value 1.
Question 4 · 3 marks · Solve
|
“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.
Question 4 · mark scheme
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
▸ Model answer. x = 197.5° and x = 342.5°
Question 5 · 2 marks · Explain
|
“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.
Question 5 · mark scheme
2 marks available. Award a mark for each point made.
▸ Maximum of sine is 1. M1
▸ Conclusion. C1
▸ Model answer. The sine graph never goes above 1 or below −1, so there is no angle with sine 1.4.
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.” |
HOW TO ANSWER IT Command word: Write down. Worth 2 marks, so plan before writing.
Question 6 · mark scheme
2 marks available. Award a mark for each point made.
▸ 180 − 40. M1
▸ 140. A1
▸ Model answer. 140°