IGCSE · Practice questions · Graphs of sine, cosine and tangent

Read the wave.

Extended. Six original Cambridge-style questions on the graphs of y = sin x, y = cos x and y = tan x.

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

The method

Sketch the graph first, then read across.

Sine: starts at 0, peaks at \((90^\circ, 1)\), lowest at \((270^\circ, -1)\).
Cosine: starts at 1, lowest at \((180^\circ, -1)\), back to 1 at \(360^\circ\).
Tangent: zeros at \(0^\circ, 180^\circ, 360^\circ\); asymptotes at \(90^\circ, 270^\circ\); period \(180^\circ\).
Second angle: \(180^\circ - \alpha\) (sin), \(360^\circ - \alpha\) (cos), \(180^\circ + \alpha\) (tan).
Count solutions: draw the horizontal line and count the crossings.

Recall

[3 marks]

For the graph of \(y = \cos x\), \(0^\circ \le x \le 360^\circ\), write down (a) the period (b) the coordinates of the minimum point (c) the values of \(x\) where the graph crosses the \(x\)-axis.

(a) \(360^\circ\) ✓
(b) \((180^\circ, -1)\) ✓
(c) \(x = 90^\circ\) and \(x = 270^\circ\) ✓

Recall

[3 marks]

Sketch the graph of \(y = \tan x\) for \(0^\circ \le x \le 360^\circ\). Mark the asymptotes and the points where the graph crosses the \(x\)-axis.

Crosses the \(x\)-axis at \((0^\circ, 0)\), \((180^\circ, 0)\) and \((360^\circ, 0)\). ✓
Vertical asymptotes at \(x = 90^\circ\) and \(x = 270^\circ\). ✓
From \(0^\circ\) the curve rises through \((45^\circ, 1)\) towards the asymptote at \(90^\circ\); it reappears from far below, rises through \((180^\circ, 0)\) and \((225^\circ, 1)\) towards \(270^\circ\); then rises from far below to \((360^\circ, 0)\). ✓

Calculation

[2 marks]

Given that \(\sin 38^\circ = 0.616\), find the two values of \(x\) between \(0^\circ\) and \(360^\circ\) for which \(\sin x = -0.616\).

Sine is negative for \(180^\circ < x < 360^\circ\). ✓
\(x = 180^\circ + 38^\circ = 218^\circ\) and \(x = 360^\circ - 38^\circ = 322^\circ\) ✓

Analysis

[3 marks]

Without solving them, state how many solutions each equation has for \(0^\circ \le x \le 360^\circ\). Give a reason each time.

(a) \(\cos x = 0.3\) (b) \(\sin x = 1.2\) (c) \(\sin x = 1\)

(a) 2: the line \(y = 0.3\) crosses the cosine wave once before \(90^\circ\) and once after \(270^\circ\). ✓
(b) 0: \(\sin x\) is never greater than 1. ✓
(c) 1: \(y = 1\) touches the sine graph only at its maximum, \(x = 90^\circ\). ✓

Calculation

[2 marks]

Given that \(\tan 72^\circ = 3.08\), find the other value of \(x\) between \(0^\circ\) and \(360^\circ\) for which \(\tan x = 3.08\), and one value for which \(\tan x = -3.08\).

Period \(180^\circ\): \(72^\circ + 180^\circ = 252^\circ\). ✓
\(\tan x\) is negative between \(90^\circ\) and \(180^\circ\): \(180^\circ - 72^\circ = 108^\circ\) (or \(360^\circ - 72^\circ = 288^\circ\)). ✓

Analysis

[4 marks]

A seat on a big wheel is \(h\) metres above the ground after the wheel has turned through \(x^\circ\), where \(h = 10 - 8\cos x\).

(a) Find the height of the seat when \(x = 0^\circ\).

(b) Use your knowledge of the cosine graph to find the greatest height of the seat and the value of \(x\) at which it occurs.

(a) \(h = 10 - 8 \times 1 = 2\) m ✓
(b) \(h\) is greatest when \(\cos x\) is least, \(\cos x = -1\) at \(x = 180^\circ\): \(h = 10 + 8 = 18\) m. ✓
(c) \(14 = 10 - 8\cos x\) gives \(\cos x = -0.5\). ✓
\(\cos 60^\circ = 0.5\), so cosine is \(-0.5\) at \(180^\circ - 60^\circ = 120^\circ\) and \(180^\circ + 60^\circ = 240^\circ\). ✓

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.