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AS & A Level · Practice questions · Graphs of trigonometric functions

Sketch the wave.

Six original Cambridge-style questions on the graphs of sine, cosine and tangent: amplitude, period, asymptotes, finding constants and a tide model.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
The method

Basic shape first, then one transformation at a time.

01
Recall
[3 marks]

(a) State the period and the range of \(y = 5\sin\dfrac{x}{2}\).

(b) State the period of \(y = 2\tan 3x\).

  • (a) Period \(\dfrac{2\pi}{1/2} = 4\pi\); range \(-5 \le y \le 5\). ✓
  • (b) Period \(\dfrac{\pi}{3}\). ✓
02
Calculation
[4 marks]

Sketch the graph of \(y = 1 - 2\cos x\) for \(0 \le x \le 2\pi\). State the coordinates of the maximum and minimum points and of the points where the graph crosses the axes.

  • Reflect \(y = \cos x\) in the \(x\)-axis, stretch by 2, move up 1: range \(-1 \le y \le 3\). ✓
  • Minima \((0, -1)\) and \((2\pi, -1)\); maximum \((\pi, 3)\). The \(y\)-intercept is \((0, -1)\). ✓
  • \(x\)-intercepts: \(1 - 2\cos x = 0\), \(\cos x = \tfrac{1}{2}\), so \(\left(\dfrac{\pi}{3}, 0\right)\) and \(\left(\dfrac{5\pi}{3}, 0\right)\). ✓
03
Calculation
[3 marks]

For the graph of \(y = \tan\left(x - \dfrac{\pi}{6}\right)\), \(0 \le x \le \pi\), find the equation of the asymptote and the exact coordinates of the points where the graph crosses the axes.

  • Asymptote: \(x - \dfrac{\pi}{6} = \dfrac{\pi}{2}\), so \(x = \dfrac{2\pi}{3}\). ✓
  • \(x\)-axis: \(x - \dfrac{\pi}{6} = 0\), so \(\left(\dfrac{\pi}{6}, 0\right)\). ✓
  • \(y\)-axis: \(y = \tan\left(-\dfrac{\pi}{6}\right) = -\dfrac{1}{\sqrt{3}} = -\dfrac{\sqrt{3}}{3}\), so \(\left(0, -\dfrac{\sqrt{3}}{3}\right)\). ✓
04
Analysis
[4 marks]

The graph of \(y = a + b\cos 2x\) passes through the points \((0, 7)\) and \(\left(\dfrac{\pi}{2}, -1\right)\).

(a) Find \(a\) and \(b\).

(b) State the range of \(y\).

(c) Find the values of \(x\) in \(0 \le x \le \pi\) for which \(y = 3\).

  • (a) At \(x = 0\): \(a + b = 7\). At \(x = \dfrac{\pi}{2}\): \(\cos\pi = -1\), so \(a - b = -1\). Hence \(a = 3\), \(b = 4\). ✓
  • (b) \(-1 \le y \le 7\) ✓
  • (c) \(3 + 4\cos 2x = 3\) gives \(\cos 2x = 0\); \(0 \le 2x \le 2\pi\), so \(2x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}\). ✓
  • \(x = \dfrac{\pi}{4}\) or \(x = \dfrac{3\pi}{4}\) ✓
05
Analysis
[3 marks]

Find the set of values of \(k\) for which the equation \(2 - 3\sin x = k\) has exactly two solutions for \(0 \le x \le \pi\).

  • For \(0 \le x \le \pi\), \(\sin x\) rises from 0 to 1 and back to 0, so \(y = 2 - 3\sin x\) falls from 2 to \(-1\) at \(x = \dfrac{\pi}{2}\) and rises back to 2. ✓
  • A horizontal line \(y = k\) meets this twice when \(-1 < k \le 2\); at \(k = -1\) it touches once. ✓
  • Answer: \(-1 < k \le 2\) ✓
06
Analysis
[5 marks]

The depth of water in a harbour \(t\) hours after midnight is modelled by \(d = 6 + 2.5\cos\left(\dfrac{\pi t}{6}\right)\) metres, for \(0 \le t \le 24\).

(a) State the greatest and least depths and the times at which they occur.

(b) A ship needs at least 7.25 m of water. For how many hours in the 24-hour period can it enter the harbour?

  • (a) Period \(\dfrac{2\pi}{\pi/6} = 12\) hours. Greatest depth 8.5 m at \(t = 0, 12, 24\); least depth 3.5 m at \(t = 6, 18\). ✓
  • (b) \(6 + 2.5\cos\dfrac{\pi t}{6} \ge 7.25\) gives \(\cos\dfrac{\pi t}{6} \ge 0.5\). ✓
  • In one cycle \(0 \le \dfrac{\pi t}{6} \le 2\pi\): \(\dfrac{\pi t}{6} \le \dfrac{\pi}{3}\) or \(\dfrac{\pi t}{6} \ge \dfrac{5\pi}{3}\), i.e. \(t \le 2\) or \(t \ge 10\). ✓
  • Over 24 hours: \(0 \le t \le 2\), \(10 \le t \le 14\), \(22 \le t \le 24\). ✓
  • Total \(2 + 4 + 2 = 8\) hours ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.