EDEXCEL GCSE MATHS · HIGHER
Graph of the cosine function
More trigonometry · Lesson 3 of 9
Warm-up
Answer each one, then check.
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
Learning Objectives
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.
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°.
The Cosine Graph
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Cosine x equals 0.5 has two solutions: 60 degrees and 300 degrees. |
The graph of y equals cosine x from 0 to 360 degrees, showing the maximum at 0 and 360, the minimum at 180 and the two solutions of cosine x equals 0.5.
Key Values of Cosine
These are worth learning.
|
x |
0° |
60° |
90° |
180° |
|---|---|---|---|---|
|
cos x |
1 |
0.5 |
0 |
−1 | 0 | 0.5 | 1 |
Features of the Graph
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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°. |
|
Finding Two Solutions
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° |
Solving cos x = 0.5
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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°
Solving a Negative Value
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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°
Sine and Cosine Graphs
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Y = SIN X |
Y = COS X |
|
▸ Starts at 0. ▸ Maximum at 90°, minimum at 270°. ▸ Second solution: 180°− x. |
▸ Starts at 1. ▸ Maximum at 0° and 360°, minimum at 180°. ▸ Second solution: 360°− x. |
Key Terms
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Cosine graph The wave y = cos x. |
Period 360° for both sine and cosine. |
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Inverse cosine cos ⁻¹, the calculator function that finds an angle. |
Translation Sliding a graph without turning it. |
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Symmetry The graph mirrors itself about a line. |
Solution A value of x that gives the required y. |
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. |
A good answer shows: (a) At x = 45° and 225°. (b) A translation of 90° to the left.
Can I...?
☐ 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.
Summary
✓ 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.
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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. |