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Maths · More trigonometry

Graph of the sine function

Sketch and read the graph of \(y = \sin x\), use its symmetry to find all solutions to \(\sin x = k\) between \(0^\circ\) and \(360^\circ\), and know key values.

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  • 6 key terms
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Teacher resources

The teacher copies: slides with the questions built in, the answers, and anything else attached to this lesson for whoever is teaching it.

Student handouts

The same files the students see, to print or hand out.

Warm-up

Answer each one, then check.

  1. 1

    What is \(\sin 30^\circ\)?

    Show answerHide answer

    \(0.5\)

  2. 2

    What is \(\sin 90^\circ\)?

    Show answerHide answer

    \(1\)

  3. 3

    What is the largest value \(\sin x\) can take?

    Show answerHide answer

    \(1\)

  4. 4

    What does SOH mean?

    Show answerHide answer

    \(\sin = \text{opposite} \div \text{hypotenuse}\)

  5. 5

    Which mode must a calculator be in for degrees?

    Show answerHide answer

    DEG

Learning Objectives

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

SINE GRAPH

The graph of \(y = \sin x\) is a smooth wave that repeats every \(360^\circ\).

It starts at 0, rises to 1 at \(90^\circ\), returns to 0 at \(180^\circ\), falls to \(-1\) at \(270^\circ\) and returns to 0 at \(360^\circ\).

Key Values of Sine

These are worth learning.

  • \(\sin x\)

    \(0^\circ\): \(0\). \(30^\circ\): \(0.5\). \(90^\circ\): \(1\). \(150^\circ\): \(0.5\) | \(0\) | \(-1\) | \(0\)

Features of the Graph

  • Period

    Repeats every \(360^\circ\).

  • Maximum

    \(1\) at \(x = 90^\circ\).

  • Minimum

    \(-1\) at \(x = 270^\circ\).

  • Roots

    \(x = 0^\circ,\ 180^\circ,\ 360^\circ\).

  • Symmetry

    Symmetrical about \(x = 90^\circ\) and \(x = 270^\circ\).

Finding Two Solutions

Use symmetry about \(x = 90^\circ\).

  1. 1 Use the calculator

    Find \(x = \sin^{-1}(k)\); this is the first solution
  2. 2 Reflect about 90

    Second solution \(= 180^\circ - x\)
  3. 3 Check the range

    Both must be between \(0^\circ\) and \(360^\circ\)
  4. 4 Negative values

    If \(k < 0\), the solutions are between \(180^\circ\) and \(360^\circ\)

Solving sin x = 0.5

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

Show the solutionHide the solution
  1. 1 Calculator \(x = \sin^{-1}(0.5) = 30^\circ\)
  2. 2 Reflect \(x = 180^\circ - 30^\circ = 150^\circ\)

Answer\(x = 30^\circ\) and \(x = 150^\circ\)

Solving a Negative Value

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

Show the solutionHide the solution
  1. 1 Calculator gives \(\sin^{-1}(-0.5) = -30^\circ\), so use the positive \(30^\circ\) as a reference
  2. 2 Below the axis Solutions are \(180^\circ + 30^\circ\) and \(360^\circ - 30^\circ\)
  3. 3 Values \(210^\circ\) and \(330^\circ\)

Answer\(x = 210^\circ\) and \(x = 330^\circ\)

Sketch and Solve

(a) Sketch \(y = \sin x\) for \(0^\circ\) to \(360^\circ\). (b) Mark \(y = 0.8\) and read the two solutions from your sketch. (c) Calculate them: \(\sin^{-1}(0.8) = 53.1^\circ\).

1. Draw the axes with \(90^\circ\) marks.

2. Use symmetry about \(90^\circ\).

A good answer shows: (b) About \(53^\circ\) and \(127^\circ\). (c) \(x = 53.1^\circ\) and \(180^\circ - 53.1^\circ = 126.9^\circ\).

Can I...?

  1. 1Sketch the sine graph.
  2. 2State its maximum and minimum.
  3. 3State its period.
  4. 4Recall sin 30 and sin 90.
  5. 5Use sin inverse.
  6. 6Find the second solution.
  7. 7Solve for negative values.
  8. 8Read solutions from a graph.

Summary & Exam Focus

  • Sine repeats every \(360^\circ\), max 1, min \(-1\).
  • \(\sin x = k\) has two solutions in \(0^\circ\) to \(360^\circ\) for \(-1 < k < 1\).
  • Positive \(k\): \(x\) and \(180^\circ - x\).
  • Negative \(k\): angles in the second half.

Exam focus

Solve \(\sin x = 0.6\) for \(0^\circ \le x \le 360^\circ\). Give answers correct to 1 decimal place. (3 marks) (3 marks)

Second solution is \(180^\circ - x\). Check both are in range.

Key terms

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

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.

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 \(\sin x = 0.6\) for \(0^\circ \le x \le 360^\circ\). Give your answers correct to 1 decimal place.

Mark scheme — 3 marks available

  • \(\sin^{-1}(0.6) = 36.9^\circ\) — B1
  • \(180 - 36.9\) — M1
  • 143.1 — A1

Model answer

\(x = 36.9^\circ\) and \(x = 143.1^\circ\)

2. Exam question Write down 2 marks Easier

Write down the value of (a) \(\sin 90^\circ\) (b) \(\sin 270^\circ\)

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 \(y = \sin x\) for \(0^\circ \le x \le 360^\circ\). Use the graph to solve \(\sin x = 0.5\) and to write down the maximum value of \(\sin x\).

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

Mark scheme — 3 marks available

  • 30 — B1
  • 150 — B1
  • 1 — B1

Model answer

\(x = 30^\circ\) and \(x = 150^\circ\); maximum value \(1\).

4. Exam question Solve 3 marks Easier

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

Mark scheme — 3 marks available

  • Reference angle \(17.5^\circ\) — B1
  • \(180 + 17.5\) or \(360 - 17.5\) — M1
  • Both answers — A1

Model answer

\(x = 197.5^\circ\) and \(x = 342.5^\circ\)

5. Exam question Explain 2 marks Easier

Amir says that \(\sin x = 1.4\) has a solution. Explain why he is wrong.

Mark scheme — 2 marks available

  • 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.

6. Exam question Write down 2 marks Easier

\(\sin 40^\circ = 0.64\) (to 2 decimal places). Write down another angle between \(0^\circ\) and \(360^\circ\) with sine 0.64.

Mark scheme — 2 marks available

  • \(180 - 40\) — M1
  • 140 — A1

Model answer

\(140^\circ\)

7. Multiple choice 1 mark Easier

What is the maximum value of \(\sin x\)?

  1. A 0
  2. B 90
  3. C 1 Correct
  4. D 360

Why: The graph rises to 1 at 90 degrees.

8. Multiple choice 1 mark Core

\(\sin 30^\circ = 0.5\). Another solution between 0 and 360 is...

  1. A 60
  2. B 150 Correct
  3. C 210
  4. D 330

Why: \(180 - 30 = 150\).

9. Multiple choice 1 mark Core

The period of \(y = \sin x\) is...

  1. A 90
  2. B 180
  3. C 270
  4. D 360 Correct

Why: The wave repeats every 360 degrees.

10. Multiple choice 1 mark Core

\(\sin x = -1\) at \(x =\)...

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

Why: The minimum of sine is at 270 degrees.

11. Multiple choice 1 mark Core

How many solutions does \(\sin x = 0.2\) have for \(0 \le x \le 360\)?

  1. A 1
  2. B 2 Correct
  3. C 3
  4. D 4

Why: Two: one on each side of 90 degrees.

12. Multiple choice 1 mark Stretch

\(\sin x = -0.4\) has solutions in which range?

  1. A 0 to 90
  2. B 90 to 180
  3. C 180 to 360 Correct
  4. D All of them

Why: Negative sine values occur between 180 and 360 degrees.