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Maths · More trigonometry

Graph of the cosine function

Sketch and read the graph of \(y = \cos x\), use its symmetry to find all solutions to \(\cos x = k\) between \(0^\circ\) and \(360^\circ\), and compare it with the sine graph.

Higher
6 key terms
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Warm-up

Answer each one, then check.

1

What is \(\cos 60^\circ\)?

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\(0.5\)
2

What is \(\cos 0^\circ\)?

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\(1\)
3

What does CAH mean?

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\(\cos = \text{adjacent} \div \text{hypotenuse}\)
4

What is the period of the sine graph?

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\(360^\circ\)
5

What are the largest and smallest values of sine?

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\(1\) and \(-1\)

Learning Objectives

1Sketch the graph of \(y = \cos x\) for \(0^\circ \le x \le 360^\circ\).
2Recall key values of cosine.
3Use symmetry to find a second solution of \(\cos x = k\).
4Describe how the cosine and sine graphs are related.

COSINE GRAPH

The graph of \(y = \cos x\) is the sine wave shifted \(90^\circ\) to the left.

It starts at 1 when \(x = 0^\circ\), falls to 0 at \(90^\circ\), reaches \(-1\) at \(180^\circ\), is 0 at \(270^\circ\) and back to 1 at \(360^\circ\).

The Cosine Graph

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.

\(\cos x\)
\(0^\circ\): \(1\). \(60^\circ\): \(0.5\). \(90^\circ\): \(0\). \(180^\circ\): \(-1\) | \(0\) | \(0.5\) | \(1\)

Features of the Graph

Period
Repeats every \(360^\circ\).
Maximum
\(1\) at \(x = 0^\circ\) and \(360^\circ\).
Minimum
\(-1\) at \(x = 180^\circ\).
Roots
\(x = 90^\circ\) and \(270^\circ\).
Symmetry
Symmetrical about \(x = 180^\circ\).

Finding Two Solutions

Use symmetry about \(x = 180^\circ\).

1 Use the calculator
Find \(x = \cos^{-1}(k)\); this is the first solution
2 Reflect about 180
Second solution \(= 360^\circ - x\)
3 Check the range
Both must be between \(0^\circ\) and \(360^\circ\)
4 Negative values
\(\cos^{-1}\) of a negative gives an angle between \(90^\circ\) and \(180^\circ\)

Solving cos x = 0.5

Solve \(\cos x = 0.5\) for \(0^\circ \le x \le 360^\circ\).

Show the solutionHide the solution

1 Calculator \(x = \cos^{-1}(0.5) = 60^\circ\)
2 Reflect \(x = 360^\circ - 60^\circ = 300^\circ\)

Answer\(x = 60^\circ\) and \(x = 300^\circ\)

Solving a Negative Value

Solve \(\cos x = -0.7\) for \(0^\circ \le x \le 360^\circ\), correct to 1 decimal place.

Show the solutionHide the solution

1 Calculator \(x = \cos^{-1}(-0.7) = 134.4^\circ\)
2 Reflect \(360^\circ - 134.4^\circ = 225.6^\circ\)

Answer\(x = 134.4^\circ\) and \(x = 225.6^\circ\)

Sine and Cosine Graphs

y = sin x

Starts at 0.
Maximum at \(90^\circ\), minimum at \(270^\circ\).
Second solution: \(180^\circ - x\).

y = cos x

Starts at 1.
Maximum at \(0^\circ\) and \(360^\circ\), minimum at \(180^\circ\).
Second solution: \(360^\circ - x\).

Sine Meets Cosine

Sketch \(y = \sin x\) and \(y = \cos x\) on the same axes for \(0^\circ\) to \(360^\circ\). (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 \(x = 45^\circ\) and \(225^\circ\). (b) A translation of \(90^\circ\) to the left.

Can I...?

1Sketch the cosine graph.
2State its maximum and minimum.
3Recall cos 0, cos 60 and cos 90.
4Use cos inverse.
5Find the second solution.
6Solve for negative values.
7Compare sine and cosine.
8Read solutions from a graph.

Summary & Exam Focus

Cosine starts at 1, falls to \(-1\) at \(180^\circ\), returns to 1 at \(360^\circ\).
\(\cos x = k\) has two solutions in \(0^\circ\) to \(360^\circ\) for \(-1 < k < 1\).
Solutions are \(x\) and \(360^\circ - x\).
Cosine is sine shifted \(90^\circ\) left.

Exam focus

Solve \(\cos x = 0.3\) for \(0^\circ \le x \le 360^\circ\). Give answers correct to 1 decimal place. (3 marks) (3 marks)

Second solution is \(360^\circ - x\).

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Cosine graph: The wave \(y = \cos x\).
Period: \(360^\circ\) 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 \(y\).

Practice questions

Have a go at each one before you open its answer.

Question 1 Solve 3 marks

Solve \(\cos x = 0.3\) for \(0^\circ \le x \le 360^\circ\). Give your answers correct to 1 decimal place.
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Model answer

\(x = 72.5^\circ\) and \(x = 287.5^\circ\)

Mark scheme
- \(\cos^{-1}(0.3) = 72.5^\circ\) — B1
- \(360 - 72.5\) — M1
- 287.5 — A1
Question 2 Write down 2 marks

Write down the value of (a) \(\cos 0^\circ\) (b) \(\cos 180^\circ\)
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Model answer

(a) 1 (b) \(-1\)

Mark scheme
- 1 — B1
- \(-1\) — B1
Question 3 Use the graph 3 marks

The graph shows \(y = \cos x\) for \(0^\circ \le x \le 360^\circ\). Use the graph to solve \(\cos x = -0.5\) and to write down the minimum value of \(\cos x\).
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Model answer

\(x = 120^\circ\) and \(x = 240^\circ\); minimum value \(-1\).

Mark scheme
- 120 — B1
- 240 — B1
- \(-1\) — B1
Question 4 Solve 3 marks

Solve \(\cos x = -0.7\) for \(0^\circ \le x \le 360^\circ\). Give your answers correct to 1 decimal place.
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Model answer

\(x = 134.4^\circ\) and \(x = 225.6^\circ\)

Mark scheme
- \(\cos^{-1}(-0.7) = 134.4^\circ\) — B1
- \(360 - 134.4\) — M1
- 225.6 — A1
Question 5 Write down 2 marks

\(\cos 50^\circ = 0.64\) (to 2 decimal places). Write down another angle between \(0^\circ\) and \(360^\circ\) with cosine 0.64.
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Model answer

\(310^\circ\)

Mark scheme
- \(360 - 50\) — M1
- 310 — A1
Question 6 Describe 2 marks

Describe the single transformation that maps the graph of \(y = \sin x\) onto the graph of \(y = \cos x\).
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Model answer

A translation of \(90^\circ\) to the left.

Mark scheme
- Translation — M1
- \(90^\circ\) to the left — A1

Quick check

What is \(\cos 0^\circ\)?
1. A0
2. B0.5
3. C\(-1\)
4. D1
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D: 1

The cosine graph starts at its maximum of 1.
\(\cos 60^\circ = 0.5\). Another solution between 0 and 360 is...
1. A120
2. B240
3. C300
4. D330
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C: 300

\(360 - 60 = 300\).
The minimum of \(\cos x\) occurs at...
1. A\(90^\circ\)
2. B\(180^\circ\)
3. C\(270^\circ\)
4. D\(360^\circ\)
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B: \(180^\circ\)

The graph is lowest at 180 degrees.
\(\cos x = 0\) at \(x =\)...
1. A\(90^\circ\) and \(270^\circ\)
2. B\(0^\circ\) and \(180^\circ\)
3. C\(180^\circ\) and \(360^\circ\)
4. D\(45^\circ\) and \(135^\circ\)
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A: \(90^\circ\) and \(270^\circ\)

The graph crosses the axis at 90 and 270 degrees.
The cosine graph is symmetrical about...
1. A\(x = 90^\circ\)
2. B\(x = 270^\circ\)
3. C\(x = 180^\circ\)
4. D\(y = 1\)
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C: \(x = 180^\circ\)

The line \(x = 180^\circ\).
The graph of \(\cos x\) can be made from the graph of \(\sin x\) by...
1. AReflecting it in the x-axis
2. BShifting it 90 degrees right
3. CStretching it
4. DShifting it 90 degrees left
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D: Shifting it 90 degrees left

Shift the sine wave 90 degrees to the left.

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