Graphs of Trigonometric Functions

In short: The graphs of $y = sin x$ and $y = cos x$ are waves between $- 1$ and $1$ that repeat every $2 π$ (or $36 0^{\circ}$ ). For $y = a sin (b x)$ , the amplitude is $∣ a ∣$ and the period is $\frac{2 π}{b}$ , and $y = tan x$ repeats every $π$ with asymptotes.

Starting point: Years 9 and 10

Plotting y = sin x

Use your calculator to fill a table from $0^{\circ}$ to $36 0^{\circ}$ :

$x$	$0^{\circ}$	$9 0^{\circ}$	$18 0^{\circ}$	$27 0^{\circ}$	$36 0^{\circ}$
$sin x$	0	1	0	$- 1$	0
$cos x$	1	0	$- 1$	0	1

Plot and join smoothly to get two waves.

Sine starts at 0. Cosine starts at 1. Both repeat every 360°.

Words for waves

The amplitude is the height from the centre line to a peak. For $y = sin x$ it is 1.
The period is the length of one full cycle. For $y = sin x$ it is $36 0^{\circ}$ .
The centre line of $y = sin x$ is $y = 0$ .

Trig graphs

y = sin x

Starts at 0
Amplitude 1

y = cos x

Starts at 1
Sine moved left

y = tan x

Asymptotes
Period π

Changes

a: amplitude
b: period 2π/b
c: phase shift
d: centre line

Models

Tides, daylight, wheels

Year 11Graphs in radians and the tangent curve

In radians, the period of $sin x$ and $cos x$ is $2 π$ . Peaks of $sin x$ are at $x = \frac{π}{2}$ , $\frac{5 π}{2}$ , and so on.

$y = tan x$ looks different. It passes through 0 at $x = 0$ , $π$ , $2 π$ and has vertical asymptotes where $cos x = 0$ : at $x = \frac{π}{2}$ , $\frac{3 π}{2}$ . Its period is $π$ .

Changing amplitude and period

For $y = a sin (b x)$ and $y = a cos (b x)$ :

amplitude $= ∣ a ∣$
period $= \frac{2 π}{b}$

So $y = 3 sin 2 x$ has amplitude 3 and period $π$ . A negative $a$ flips the wave upside down. For $y = tan (b x)$ the period is $\frac{π}{b}$ .

The rule works when $b$ is a fraction too: $y = 2 cos \frac{x}{3}$ has $b = \frac{1}{3}$ , so its period is $\frac{2 π}{1/3} = 6 π$ . And $y = sin \frac{π x}{5}$ has $b = \frac{π}{5}$ , so its period is 10.

Range. $sin$ and $cos$ run from $- 1$ to $1$ , so $y = a sin (\dots) + d$ runs from $d - ∣ a ∣$ to $d + ∣ a ∣$ . For $y = 3 + 2 cos 4 x$ the range is $1 \leq y \leq 5$ . The inside of the bracket does not change the range.

Year 12The general wave

$y = a sin (b (x - c)) + d$

$∣ a ∣$ = amplitude
$\frac{2 π}{b}$ = period
$c$ = phase shift (horizontal translation right)
$d$ = centre line (vertical translation)

Example: Sofia's Ferris wheel. The wheel has diameter 40 m, its lowest point is 5 m above the ground, and one turn takes 12 minutes. The centre line is $5 + 20 = 25$ m and the amplitude is 20 m. Starting at the bottom, a negative cosine fits:

$h (t) = - 20 cos (\frac{π}{6} t) + 25$

because $b = \frac{2 π}{12} = \frac{π}{6}$ . At $t = 6$ minutes, $h = - 20 cos π + 25 = 45$ m, the top.

Solving with graphs. The number of solutions of $sin x = 0.4$ for $0 \leq x \leq 2 π$ is the number of times the line $y = 0.4$ crosses the wave: 2.

Finding the equation from the graph. The centre line is halfway between the maximum and minimum, and the amplitude is half the gap. A wave with maximum 7 and minimum $- 1$ has $d = 3$ and $a = 4$ . If it is at its maximum when $x = 0$ with period $2 π$ , it is $y = 4 cos x + 3$ .

Solving a model. For $T = 20 + 6 sin \frac{π t}{12}$ , when is $T$ above 23? Solve $sin \frac{π t}{12} = \frac{1}{2}$ : $\frac{π t}{12} = \frac{π}{6}$ or $\frac{5 π}{6}$ , so $t = 2$ or $t = 10$ . Between them the temperature is above 23, for 8 hours. The time between a maximum and the next minimum is always half a period.

Fastest change. A wave rises or falls fastest as it crosses its centre line, and changes slowest at a peak or trough. In calculus, the rate of change is the derivative, so the fastest rise is where the derivative is greatest.

Quick summary

$y = sin x$ and $y = cos x$ are waves between $- 1$ and $1$ with period $36 0^{\circ}$ , or $2 π$ .
Cosine is sine moved left by $9 0^{\circ}$ .
$y = tan x$ has period $π$ and asymptotes where $cos x = 0$ .
$y = a sin (b x)$ : amplitude $∣ a ∣$ , period $\frac{2 π}{b}$ .
$y = a sin (b (x - c)) + d$ : $c$ shifts right, $d$ moves the centre line.

Quiz

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Try 3 questions now

What is the value of $sin 9 0^{\circ}$ ?
1. A 0
2. B 1
3. C $- 1$
4. D 0.5
Show answer
B: 1
The sine wave reaches its peak at $9 0^{\circ}$ .
What is the value of $cos 18 0^{\circ}$ ?
1. A 0
2. B 1
3. C $- 1$
4. D 0.5
Show answer
C: $- 1$
The cosine wave is at its lowest at $18 0^{\circ}$ .
What is the amplitude of $y = sin x$ ?
1. A 1
2. B 2
3. C $36 0^{\circ}$
4. D 0
Show answer
A: 1
Centre line 0, peak 1.