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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 minus 0.5 crossing the curve twice.
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    Model answer

    \(\cos 60^\circ = 0.5\), and cosine is negative between \(90^\circ\) and \(270^\circ\), so \(x = 180 - 60 = 120^\circ\) or \(x = 180 + 60 = 240^\circ\).

    Mark scheme

    • \(120\) — M1
    • \(240\) — A1
  2. 2 Write down [3 marks]

    The diagram shows the graphs of two curves, \(P\) and \(Q\), for \(0^\circ \le x \le 360^\circ\). One is \(y = \sin x\) and the other is \(y = \cos x\). (a) State which curve is \(y = \sin x\). [1 mark] (b) Solve \(\sin x = \cos x\) for \(0^\circ \le x \le 360^\circ\). [2 marks]

    Two wave graphs, one starting at the origin and one starting at 1, crossing at two points.
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    Model answer

    (a) \(P\), because it passes through the origin. (b) The curves cross at \(x = 45^\circ\), where \(\sin 45^\circ = \cos 45^\circ = \dfrac{\sqrt{2}}{2}\), and at \(x = 225^\circ\).

    Mark scheme

    • (a) Curve P, because it starts at the origin — B1
    • (b) \(45\) — B1
    • (b) \(225\) — B1
  3. 3 Write down [3 marks]

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

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    Model answer

    (a) \(\dfrac{\sqrt{2}}{2}\). (b) \(-\dfrac{\sqrt{2}}{2}\). (c) \(-\dfrac{\sqrt{3}}{3}\), which is the same as \(-\dfrac{1}{\sqrt{3}}\).

    Mark scheme

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

    Solve \(2\cos x + 1 = 0\) for \(0^\circ \le x \le 360^\circ\). [3 marks]

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    Model answer

    \(\cos x = -\dfrac{1}{2}\), so \(x = 120^\circ\) or \(x = 240^\circ\).

    Mark scheme

    • \(\cos x = -\dfrac{1}{2}\) — M1
    • \(120\) — A1
    • \(240\) — A1
  5. 5 Solve [3 marks]

    Solve \(\tan x = -\sqrt{3}\) for \(0^\circ \le x \le 360^\circ\). [3 marks]

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    Model answer

    \(\tan 60^\circ = \sqrt{3}\), and tangent is negative between \(90^\circ\) and \(180^\circ\), and between \(270^\circ\) and \(360^\circ\). So \(x = 180 - 60 = 120^\circ\) or \(x = 360 - 60 = 300^\circ\).

    Mark scheme

    • Uses \(60^\circ\) as the related angle — M1
    • \(120\) — A1
    • \(300\) — A1
  6. 6 Write down [2 marks]

    (a) Write down the period of the graph of \(y = \tan x\). [1 mark] (b) How many solutions does \(\cos x = 0.3\) have for \(0^\circ \le x \le 720^\circ\)? [1 mark]

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    Model answer

    (a) \(180^\circ\). (b) Four solutions, two in each \(360^\circ\).

    Mark scheme

    • (a) \(180^\circ\) — B1
    • (b) 4 — 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.