5.3 The sine, cosine and tangent functions: graphs and periodicity — Learn

Prerequisites: coordinates on the unit circle; radian measure; the shapes of the sine and cosine curves from GCSE; reading a graph.

By the end of this section you can: sketch the graphs of sine, cosine and tangent; state their period and symmetry; read an amplitude, a period and a phase shift from a transformed wave; write down the range of each function.

1. The three graphs

The three functions are defined for an angle $x$ . It is easiest to read them from the unit circle. The point at angle $x$ has coordinates $(\cos x, \sin x)$ , so the height of the point is $\sin x$ and its horizontal distance from the centre is $\cos x$ .

The graph of $y = \sin x$ starts at $0$ , rises to a peak of $1$ at $x = \dfrac{\pi}{2}$ , returns to $0$ at $\pi$ , falls to a trough of $-1$ at $\dfrac{3\pi}{2}$ , and returns to $0$ at $2\pi$ . The pattern then repeats.

The graph of $y = \cos x$ has the same shape, shifted left by $\dfrac{\pi}{2}$ . It starts at $1$ , falls to $0$ at $\dfrac{\pi}{2}$ , reaches $-1$ at $\pi$ , and rises back to $1$ at $2\pi$ .

Both graphs repeat every $2\pi$ radians. The period of each is $2\pi$ .

The graph of $y = \tan x$ is different. It has a period of $\pi$ , not $2\pi$ , because $\tan x = \dfrac{\sin x}{\cos x}$ and both sine and cosine change sign together. The tangent has vertical asymptotes where $\cos x = 0$ , at $x = \dfrac{\pi}{2}$ and $x = \dfrac{3\pi}{2}$ .

Function	Period	Range	Symmetry
$y = \sin x$	$2\pi$	$-1 \le y \le 1$	odd: $\sin(-x) = -\sin x$
$y = \cos x$	$2\pi$	$-1 \le y \le 1$	even: $\cos(-x) = \cos x$
$y = \tan x$	$\pi$	all real $y$	odd: $\tan(-x) = -\tan x$

An odd function is unchanged by a half turn about the origin. An even function is unchanged by a reflection in the $y$ -axis.

2. Transforming the wave

A wave is often written in the form

y = A\sin(Bx + C) + D.

Each letter does one job.

$A$ is the amplitude, the distance from the middle line to a peak. The graph is stretched vertically by a factor $|A|$ .
$B$ changes the period. The period is $\dfrac{2\pi}{B}$ .
$C$ is the phase. The graph is shifted horizontally by $-\dfrac{C}{B}$ .
$D$ is the vertical shift. The middle line of the wave is $y = D$ .

The maximum value is $D + |A|$ and the minimum is $D - |A|$ .

Example 1. For $y = 3\sin\left(2x\right) - 1$ , state the amplitude, the period, and the maximum and minimum values.

The amplitude is $3$ . The period is $\dfrac{2\pi}{2} = \pi$ . The middle line is $y = -1$ , so the maximum is $-1 + 3 = 2$ and the minimum is $-1 - 3 = -4$ .

Example 2. Write down the period and the phase shift of $y = \cos\left(3x - \dfrac{\pi}{2}\right)$ .

Here $B = 3$ and $C = -\dfrac{\pi}{2}$ . The period is $\dfrac{2\pi}{3}$ . The phase shift is $-\dfrac{C}{B} = \dfrac{\pi}{6}$ , so the graph is shifted $\dfrac{\pi}{6}$ to the right.

3. Try it: the wave explorer

Predict first. You will change the amplitude $A$ , the wave number $B$ , the phase $C$ and the vertical shift $D$ in $y = A\sin(Bx + C) + D$ . Before you touch the controls, predict what happens to the number of complete waves when $B$ increases, and what happens to the peak height when $A$ is negative. Write both predictions down.

Test. Move each slider in turn and watch the curve. Check that the readout period matches the distance between two peaks on the graph. Then set $A = 1$ and $B = 2$ and count the complete waves between $x = 0$ and $x = 2\pi$ .

Explain. Write two sentences. The first explains why doubling $B$ halves the period. The second explains why a negative $A$ turns the wave upside down rather than moving it.

Watch out. Three mistakes are common with these graphs.

Using degrees on the horizontal axis. The formulas in this chapter use radians. The period of $\sin x$ is $2\pi$ , not $360$ . Write the axis in multiples of $\pi$ unless a question asks for degrees.

Reading the period as $\dfrac{B}{2\pi}$ . The period is $\dfrac{2\pi}{B}$ . A larger $B$ squeezes more waves into the same width, so a larger $B$ gives a smaller period. Test it with $B = 2$ : the period must be $\pi$ , not $4\pi$ .

Forgetting the vertical shift in the range. For $y = A\sin(Bx + C) + D$ the range runs from $D - |A|$ to $D + |A|$ . The middle line is $y = D$ , not the $x$ -axis. Re-check: find the maximum and minimum and confirm that they are equally far above and below the middle line.

You should now be able to

sketch $y = \sin x$ , $y = \cos x$ and $y = \tan x$ over one period;
state the period and range of each function;
use the odd and even symmetry of sine, cosine and tangent;
find the amplitude, period, phase shift and vertical shift of $y = A\sin(Bx + C) + D$ ;
state the maximum and minimum of a transformed wave.