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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.

  • Higher
  • 6 key terms
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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.

Practice questions

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

  1. Question 1 Solve 3 marks

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

    Show answerHide answer

    Model answer

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

    Mark scheme

    • \(\sin^{-1}(0.6) = 36.9^\circ\) — B1
    • \(180 - 36.9\) — M1
    • 143.1 — A1
  2. Question 2 Write down 2 marks

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

    Show answerHide answer

    Model answer

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

    Mark scheme

    • 1 — B1
    • \(-1\) — B1
  3. Question 3 Use the graph 3 marks

    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.
    Show answerHide answer

    Model answer

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

    Mark scheme

    • 30 — B1
    • 150 — B1
    • 1 — B1
  4. Question 4 Solve 3 marks

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

    Show answerHide answer

    Model answer

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

    Mark scheme

    • Reference angle \(17.5^\circ\) — B1
    • \(180 + 17.5\) or \(360 - 17.5\) — M1
    • Both answers — A1
  5. Question 5 Explain 2 marks

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

    Show answerHide answer

    Model answer

    The sine graph never goes above 1 or below \(-1\), so there is no angle with sine 1.4.

    Mark scheme

    • Maximum of sine is 1 — M1
    • Conclusion — C1
  6. Question 6 Write down 2 marks

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

    Show answerHide answer

    Model answer

    \(140^\circ\)

    Mark scheme

    • \(180 - 40\) — M1
    • 140 — A1

Quick check

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

    1. A0
    2. B90
    3. C1
    4. D360
    Show answerHide answer

    C: 1

    The graph rises to 1 at 90 degrees.

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

    1. A60
    2. B150
    3. C210
    4. D330
    Show answerHide answer

    B: 150

    \(180 - 30 = 150\).

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

    1. A90
    2. B180
    3. C270
    4. D360
    Show answerHide answer

    D: 360

    The wave repeats every 360 degrees.

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

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

    A: \(270^\circ\)

    The minimum of sine is at 270 degrees.

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

    1. A1
    2. B2
    3. C3
    4. D4
    Show answerHide answer

    B: 2

    Two: one on each side of 90 degrees.

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

    1. A0 to 90
    2. B90 to 180
    3. C180 to 360
    4. DAll of them
    Show answerHide answer

    C: 180 to 360

    Negative sine values occur between 180 and 360 degrees.

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