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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.

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  • 6 key terms
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Teacher resources

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Student handouts

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Warm-up

Answer each one, then check.

  1. 1

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

    Show answerHide answer

    \(0.5\)

  2. 2

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

    Show answerHide answer

    \(1\)

  3. 3

    What does CAH mean?

    Show answerHide answer

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

  4. 4

    What is the period of the sine graph?

    Show answerHide answer

    \(360^\circ\)

  5. 5

    What are the largest and smallest values of sine?

    Show answerHide answer

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

Learning Objectives

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

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. 1 Use the calculator

    Find \(x = \cos^{-1}(k)\); this is the first solution
  2. 2 Reflect about 180

    Second solution \(= 360^\circ - x\)
  3. 3 Check the range

    Both must be between \(0^\circ\) and \(360^\circ\)
  4. 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. 1 Calculator \(x = \cos^{-1}(0.5) = 60^\circ\)
  2. 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. 1 Calculator \(x = \cos^{-1}(-0.7) = 134.4^\circ\)
  2. 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...?

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

Questions and answers

12 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Solve 3 marks Easier

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

Mark scheme — 3 marks available

  • \(\cos^{-1}(0.3) = 72.5^\circ\) — B1
  • \(360 - 72.5\) — M1
  • 287.5 — A1

Model answer

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

2. Exam question Write down 2 marks Easier

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

Mark scheme — 2 marks available

  • 1 — B1
  • \(-1\) — B1

Model answer

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

3. Exam question Use the graph 3 marks Easier

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 \(\cos x\).

The graph of cosine x from 0 to 360 degrees on a grid with a dotted line at y equals minus 0.5.

Mark scheme — 3 marks available

  • 120 — B1
  • 240 — B1
  • \(-1\) — B1

Model answer

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

4. Exam question Solve 3 marks Easier

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

Mark scheme — 3 marks available

  • \(\cos^{-1}(-0.7) = 134.4^\circ\) — B1
  • \(360 - 134.4\) — M1
  • 225.6 — A1

Model answer

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

5. Exam question Write down 2 marks Easier

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

Mark scheme — 2 marks available

  • \(360 - 50\) — M1
  • 310 — A1

Model answer

\(310^\circ\)

6. Exam question Describe 2 marks Easier

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

Mark scheme — 2 marks available

  • Translation — M1
  • \(90^\circ\) to the left — A1

Model answer

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

7. Multiple choice 1 mark Easier

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

  1. A 0
  2. B 0.5
  3. C \(-1\)
  4. D 1 Correct

Why: The cosine graph starts at its maximum of 1.

8. Multiple choice 1 mark Core

\(\cos 60^\circ = 0.5\). Another solution between 0 and 360 is...

  1. A 120
  2. B 240
  3. C 300 Correct
  4. D 330

Why: \(360 - 60 = 300\).

9. Multiple choice 1 mark Core

The minimum of \(\cos x\) occurs at...

  1. A \(90^\circ\)
  2. B \(180^\circ\) Correct
  3. C \(270^\circ\)
  4. D \(360^\circ\)

Why: The graph is lowest at 180 degrees.

10. Multiple choice 1 mark Core

\(\cos x = 0\) at \(x =\)...

  1. A \(90^\circ\) and \(270^\circ\) Correct
  2. B \(0^\circ\) and \(180^\circ\)
  3. C \(180^\circ\) and \(360^\circ\)
  4. D \(45^\circ\) and \(135^\circ\)

Why: The graph crosses the axis at 90 and 270 degrees.

11. Multiple choice 1 mark Core

The cosine graph is symmetrical about...

  1. A \(x = 90^\circ\)
  2. B \(x = 270^\circ\)
  3. C \(x = 180^\circ\) Correct
  4. D \(y = 1\)

Why: The line \(x = 180^\circ\).

12. Multiple choice 1 mark Stretch

The graph of \(\cos x\) can be made from the graph of \(\sin x\) by...

  1. A Reflecting it in the x-axis
  2. B Shifting it 90 degrees right
  3. C Stretching it
  4. D Shifting it 90 degrees left Correct

Why: Shift the sine wave 90 degrees to the left.