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Exam questions · Maths · Further Trigonometry

Trigonometric Graphs and Exact Values

  • 6 exam questions
  • 16 marks
  • 9 quick checks
  1. 1 Solve [2 marks]

    The diagram shows the graph of \(y = \cos x\) for \(0^\circ \le x \le 360^\circ\), and the line \(y = 0.5\). Solve \(\cos x = 0.5\) for \(0^\circ \le x \le 360^\circ\). (2 marks)

    The graph of cosine from 0 to 360 degrees with the line y equals 0.5 crossing the curve twice.
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    Model answer

    \(\cos 60^\circ = 0.5\), so \(x = 60^\circ\). The cosine graph is symmetrical about \(180^\circ\), so the other solution is \(360 - 60 = 300^\circ\).

    Mark scheme

    • \(60\) — M1
    • \(300\) — A1
  2. 2 Write down [3 marks]

    The diagram shows a graph for \(0^\circ \le x \le 360^\circ\). (a) Write down the equation of the graph. (1 mark) (b) Write down the coordinates of the maximum point. (1 mark) (c) Write down the values of \(x\) where the graph crosses the \(x\)-axis. (1 mark)

    A wave graph between 1 and minus 1 starting at the origin, with a maximum at 90 degrees.
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    Model answer

    (a) \(y = \sin x\). (b) \((90, 1)\). (c) \(x = 0^\circ\), \(180^\circ\) and \(360^\circ\).

    Mark scheme

    • (a) \(y = \sin x\) — B1
    • (b) \((90, 1)\) — B1
    • (c) \(0, 180, 360\) — B1
  3. 3 Write down [3 marks]

    Write down the exact value of (a) \(\sin 150^\circ\) (1 mark) (b) \(\cos 120^\circ\) (1 mark) (c) \(\tan 135^\circ\) (1 mark)

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    Model answer

    (a) \(\dfrac{1}{2}\). (b) \(-\dfrac{1}{2}\). (c) \(-1\).

    Mark scheme

    • (a) \(\dfrac{1}{2}\) — B1
    • (b) \(-\dfrac{1}{2}\) — B1
    • (c) \(-1\) — B1
  4. 4 Solve [3 marks]

    Solve \(2\sin x + \sqrt{3} = 0\) for \(0^\circ \le x \le 360^\circ\). (3 marks)

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    Model answer

    \(\sin x = -\dfrac{\sqrt{3}}{2}\). Sine is negative between \(180^\circ\) and \(360^\circ\), and \(\sin 60^\circ = \dfrac{\sqrt{3}}{2}\), so \(x = 180 + 60 = 240^\circ\) or \(x = 360 - 60 = 300^\circ\).

    Mark scheme

    • \(\sin x = -\dfrac{\sqrt{3}}{2}\) — M1
    • \(240\) — A1
    • \(300\) — A1
  5. 5 Solve [3 marks]

    Solve \(\tan x = -1\) for \(0^\circ \le x \le 360^\circ\). (3 marks)

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    Model answer

    \(\tan 45^\circ = 1\), and tangent is negative between \(90^\circ\) and \(180^\circ\), and between \(270^\circ\) and \(360^\circ\). So \(x = 180 - 45 = 135^\circ\) or \(x = 360 - 45 = 315^\circ\).

    Mark scheme

    • Uses \(45^\circ\) as the related angle — M1
    • \(135\) — A1
    • \(315\) — A1
  6. 6 Explain [2 marks]

    (a) Explain why the equation \(\sin x = \dfrac{3}{2}\) has no solutions. (1 mark) (b) How many solutions does \(\cos x = -1\) have for \(0^\circ \le x \le 360^\circ\)? (1 mark)

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    Model answer

    (a) The sine of an angle is never greater than 1. (b) One solution, \(x = 180^\circ\).

    Mark scheme

    • (a) The sine of an angle is never more than 1, or the graph does not go above 1 — C1
    • (b) 1, which is \(x = 180^\circ\) — B1

Quick check

  1. 1

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

    1. A0
    2. B1
    3. C90
    4. D\(\infty\)
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    B: 1

    The sine graph is a wave between \(-1\) and \(1\).

  2. 2

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

    1. A1
    2. B0
    3. C\(-1\)
    4. D\(\dfrac{1}{2}\)
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    A: 1

    The cosine graph starts at its maximum, 1.

  3. 3

    Where are the asymptotes of \(y = \tan x\) between \(0^\circ\) and \(360^\circ\)?

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

    The tangent is undefined at \(90^\circ\) and \(270^\circ\).

  4. 4

    How many solutions does \(\sin x = \dfrac{1}{2}\) have for \(0^\circ \le x \le 360^\circ\)?

    1. A1
    2. B3
    3. C2
    4. D4
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    C: 2

    The line \(y = \dfrac{1}{2}\) crosses the sine curve twice, at \(30^\circ\) and \(150^\circ\).

  5. 5

    If \(\sin 30^\circ = \dfrac{1}{2}\), what is the other solution of \(\sin x = \dfrac{1}{2}\) between \(0^\circ\) and \(360^\circ\)?

    1. A\(120^\circ\)
    2. B\(150^\circ\)
    3. C\(210^\circ\)
    4. D\(330^\circ\)
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    B: \(150^\circ\)

    The second solution is \(180^\circ - 30^\circ = 150^\circ\).

  6. 6

    What is the period of \(y = \tan x\)?

    1. A\(180^\circ\)
    2. B\(360^\circ\)
    3. C\(90^\circ\)
    4. D\(45^\circ\)
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    A: \(180^\circ\)

    The tangent graph repeats every \(180^\circ\).

  7. 7

    Solve \(\cos x = \dfrac{1}{2}\) for \(0^\circ \le x \le 360^\circ\).

    1. A\(60^\circ\) and \(120^\circ\)
    2. B\(30^\circ\) and \(330^\circ\)
    3. C\(60^\circ\) and \(240^\circ\)
    4. D\(60^\circ\) and \(300^\circ\)
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    D: \(60^\circ\) and \(300^\circ\)

    \(\cos 60^\circ = \dfrac{1}{2}\), and the second solution is \(360^\circ - 60^\circ = 300^\circ\).

  8. 8

    Solve \(\sin x = -\dfrac{1}{2}\) for \(0^\circ \le x \le 360^\circ\).

    1. A\(30^\circ\) and \(150^\circ\)
    2. B\(120^\circ\) and \(240^\circ\)
    3. C\(210^\circ\) and \(330^\circ\)
    4. D\(150^\circ\) and \(210^\circ\)
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    C: \(210^\circ\) and \(330^\circ\)

    Sine is negative between \(180^\circ\) and \(360^\circ\), so the solutions are \(180 + 30 = 210\) and \(360 - 30 = 330\).

  9. 9

    Which equation has no solutions?

    1. A\(\cos x = -1\)
    2. B\(\sin x = 1.5\)
    3. C\(\tan x = 5\)
    4. D\(\sin x = 0\)
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    B: \(\sin x = 1.5\)

    Sine and cosine are never greater than 1, so \(\sin x = 1.5\) has no solution.