EDEXCEL GCSE MATHS · HIGHER
Graph of the sine function
More trigonometry · Lesson 2 of 9
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
|
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
|
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. |
|
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
|
(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°.
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.
|
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. |