Fundamentals of Trigonometry

Lessons

Senior 4 · Mathematics

Fundamentals of Trigonometry

5 lessons

Lesson 4 of 5

Graphs of the trigonometric functions

25 min

Unwrapping the circle

Let the point $M$ travel round the unit circle at a steady speed, and record its height, $\sin x$ , against the angle $x$ turned so far. The record rises to 1, falls through 0 to $- 1$ , comes back, and then repeats the same shape for ever. That wave is the graph of the sine function, and the same shape appears wherever something repeats smoothly: alternating current, sound, the swing of a pendulum, the height of the tide. Electronics engineers watch such waves on an oscilloscope.

Definition

Periodic function: a function $f$ is periodic if there is a number $T \gt 0$ with $f (x + T) = f (x)$ for every $x$ in its domain. The smallest such $T$ is the period.

Amplitude: for a wave that oscillates between a maximum $M$ and a minimum $m$ , the amplitude is $\frac{M - m}{2}$ , half the distance from trough to crest.

Example: $\sin(x + 2\pi) = \sin x$ for every $x$ , because $x$ and $x + 2\pi$ are coterminal; no smaller positive shift works, so $\sin$ has period $2\pi$ . Its amplitude is $\frac{1 - (-1)}{2} = 1$ .

The sine and cosine graphs

$x$	0	$\frac{\pi}{6}$	$\frac{\pi}{2}$	$\frac{5\pi}{6}$	$π \pi$	$\frac{7\pi}{6}$	$\frac{3\pi}{2}$	$\frac{11\pi}{6}$	$2\pi$
$\sin x$	0	0.5	1	0.5	0	$- 0.5$	$- 1$	$- 0.5$	0
$\cos x$	1	0.87	0	$- 0.87$	$- 1$	$- 0.87$	0	0.87	1

Property	$y = \sin x$	$y = \cos x$
Domain, range	$\mathbb{R}$ , $[- 1, 1]$	$\mathbb{R}$ , $[- 1, 1]$
Period, amplitude	$2\pi$ , 1	$2\pi$ , 1
Zeros	$x = k\pi$	$x = \frac{\pi}{2} + k\pi$
Maximum 1 at	$x = \frac{\pi}{2} + 2k\pi$	$x = 2k\pi$
Minimum $- 1$ at	$x = \frac{3\pi}{2} + 2k\pi$	$x = \pi + 2k\pi$
Symmetry	odd: $\sin(-x) = -\sin x$ , symmetric about $O$	even: $\cos(-x) = \cos x$ , symmetric about the $y$ -axis

( $k \in \mathbb{Z}$ throughout.) Since $\cos x = \sin\left(x + \frac{\pi}{2}\right)$ , the cosine graph is the sine graph moved $\frac{\pi}{2}$ to the left.

The tangent graph

$\tan x = \frac{\sin x}{\cos x}$ is undefined where $\cos x = 0$ , that is at $x = \frac{\pi}{2} + k\pi$ . Near these values the denominator is tiny and $|\tan x|$ becomes huge, so the graph has vertical asymptotes there.

Properties: domain $\mathbb{R} \setminus \left\{\frac{\pi}{2} + k\pi\right\}$ ; range $\mathbb{R}$ ; period $π \pi$ (since $\tan(x + \pi) = \frac{-\sin x}{-\cos x} = \tan x$ ); zeros at $x = k\pi$ ; odd; increasing on each interval between asymptotes; no amplitude.

Worked example 1: reading solutions from a graph

(a) Solve $\sin x = \frac{1}{2}$ for $0 \le x \le 2\pi$ . (b) How many solutions has $\sin x = 0.3$ for $0 \le x \le 4\pi$ ?

Solution

The line $y = \frac{1}{2}$ meets one arch of the sine curve twice, symmetrically about $x = \frac{\pi}{2}$ . One solution is $\frac{\pi}{6}$ ; the other is $\pi - \frac{\pi}{6} = \frac{5\pi}{6}$ .
The line $y = 0.3$ crosses the curve twice in every period where the curve is above it, that is, twice in each of $[0, 2\pi]$ and $[2\pi, 4\pi]$ : 4 solutions.

Stretching and lifting the waves

For real numbers $a \ne 0$ , $b \gt 0$ and $c$ :

$y = a\sin x$ and $y = a\cos x$ have amplitude $∣ a ∣$ (a vertical stretch);
$y = \sin x + c$ is the sine graph lifted by $c$ : it oscillates about the line $y = c$ ;
$y = \sin bx$ has period $\frac{2\pi}{b}$ (a horizontal squeeze); you will study such periods fully in S5.

Worked example 2: sketching $y = 3\cos x - 1$

Solution The amplitude is 3 and the centre line is $y = - 1$ , so the range is $[- 1 - 3, - 1 + 3] = [- 4, 2]$ . The period is still $2\pi$ . The maximum 2 occurs where $\cos x = 1$ ( $x = 0, 2\pi$ ) and the minimum $- 4$ where $\cos x = -1$ ( $x = \pi$ ). The curve crosses its centre line at $x = \frac{\pi}{2}$ and $\frac{3\pi}{2}$ . Plot these five key points and join them with a smooth wave (graph above).

Worked example 3: a tide model

The depth of water at a jetty is modelled by $d = 5 + 2\sin\frac{\pi t}{6}$ , with $d$ in metres and $t$ in hours after midnight (illustrative). Find the period, the greatest and least depths and when they first occur, and the depth at 1 a.m.

Solution $\sin\frac{\pi t}{6}$ repeats when $\frac{\pi t}{6}$ increases by $2\pi$ , that is, when $t$ increases by 12: the period is 12 hours. The greatest depth is $5 + 2 = 7\,\mathrm{m}$ when $\frac{\pi t}{6} = \frac{\pi}{2}$ , at $t = 3$ (3 a.m.); the least is $5 - 2 = 3\,\mathrm{m}$ when $\frac{\pi t}{6} = \frac{3\pi}{2}$ , at $t = 9$ . At 1 a.m., $d = 5 + 2\sin\frac{\pi}{6} = 5 + 1 = 6\,\mathrm{m}$ .

Warning

Label the $x$ -axis in radians ( $\frac{\pi}{2}, \pi, \ldots$ ) unless told otherwise, and keep the same scale on both sides of the origin. When sketching $y = \tan x$ , draw the asymptotes first and never join the curve across them: the graph is not continuous at $x = \frac{\pi}{2}$ .

Tip

To sketch one period of a sine or cosine graph, plot five key points: the start, a quarter, a half, three quarters and the end of the period. For $y = \sin x$ they are $(0, 0), \left(\frac{\pi}{2}, 1\right), (\pi, 0), \left(\frac{3\pi}{2}, -1\right), (2\pi, 0)$ .

JensenMath: Graphing Sine and Cosine (full lesson). Watch for how the special-angle values give the points of the graphs, and for the period and amplitude.

✦

Practice

Sketch a graph whenever it helps.

State the domain, range and period of $y = \sin x$ , $y = \cos x$ and $y = \tan x$ .
Use the graphs to solve (a) $\cos x = 0$ for $0 \le x \le 2\pi$ (b) $\sin x = -1$ for $-2\pi \le x \le 2\pi$ .
How many solutions has $\sin x = 0.4$ for $0 \le x \le 4\pi$ ?
Give the amplitude, range and period of (a) $y = 4\sin x$ (b) $y = -2\cos x + 3$ (c) $y = \sin 3x$ .
Explain with the unit circle why $\cos$ is an even function and $\sin$ is odd.
Find the greatest and least values of $y = 2 - 3\sin x$ on $[0, 2\pi]$ , and where they occur.
Write the equations of the asymptotes of $y = \tan x$ for $-2\pi \le x \le 2\pi$ .
For the tide model $d = 5 + 2\sin\frac{\pi t}{6}$ , find the times in $[0, 12]$ at which the depth is 6 m.

Show answersHide answers

$\sin$ and $\cos$ : domain $\mathbb{R}$ , range $[- 1, 1]$ , period $2\pi$ . $\tan$ : domain $\mathbb{R} \setminus \left\{\frac{\pi}{2} + k\pi\right\}$ , range $\mathbb{R}$ , period $π \pi$ .
(a) $\frac{\pi}{2}$ , $\frac{3\pi}{2}$ (b) $-\frac{\pi}{2}$ , $\frac{3\pi}{2}$ .
4.
(a) 4, $[- 4, 4]$ , $2\pi$ (b) 2, $[1, 5]$ , $2\pi$ (c) 1, $[- 1, 1]$ , $\frac{2\pi}{3}$ .
The points for $x$ and $- x$ are reflections in the $x$ -axis: same $x$ -coordinate ( $\cos(-x) = \cos x$ ) and opposite $y$ -coordinates ( $\sin(-x) = -\sin x$ ).
Greatest 5 at $x = \frac{3\pi}{2}$ (where $\sin x = -1$ ); least $- 1$ at $x = \frac{\pi}{2}$ .
$x = -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}$ .
$\sin\frac{\pi t}{6} = \frac{1}{2}$ , so $\frac{\pi t}{6} = \frac{\pi}{6}$ or $\frac{5\pi}{6}$ : $t = 1$ or $t = 5$ (1 a.m. and 5 a.m.).

Quick check

What is the period of y = tan x?

Key points

$\sin$ and $\cos$ : range $[- 1, 1]$ , period $2\pi$ ; $\sin$ is odd, $\cos$ is even.
$\tan$ : period $π \pi$ , asymptotes at $x = \frac{\pi}{2} + k\pi$ .
$y = a\sin x + c$ has amplitude $∣ a ∣$ and oscillates about $y = c$ ; $y = \sin bx$ has period $\frac{2\pi}{b}$ .

Sources and credits

Written by Ganzaa: the explanations, the graphs, the tables, the worked examples (the tide model is illustrative), the practice items and the questions.
Photo: Sine wave 10 kHz displayed on analog oscilloscope, Pittigrilli, Wikimedia Commons, CC BY-SA 4.0.
Video: Graphing Sine and Cosine (full lesson), JensenMath, embedded from YouTube.
This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1Unwrapping the circle

2The sine and cosine graphs

3The tangent graph

4Stretching and lifting the waves

Unwrapping the circle

The sine and cosine graphs

The tangent graph

Stretching and lifting the waves