# Graphs of Trigonometric Functions

> The waves of sin, cos and tan. Amplitude, period, phase shift and modelling things that repeat.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/trigonometric-graphs/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The Ferris wheel
Sofia rides the big Ferris wheel at Luna Park. She starts at the bottom, rises to the top, comes back down, and does it again and again. If you plotted her height against time, you would see a smooth wave that repeats every turn.

Tides, sound, daylight hours and heartbeats all repeat like this. Sine and cosine are the functions that describe repeating things. Their graphs are the waves at the heart of physics, music and engineering.

**In short:** The graphs of $y = \sin x$ and $y = \cos x$ are waves between $-1$ and $1$ that repeat every $2\pi$ (or $360^\circ$). For $y = a\sin(bx)$, the amplitude is $|a|$ and the period is $\dfrac{2\pi}{b}$, and $y = \tan x$ repeats every $\pi$ with asymptotes.

## Starting point: Years 9 and 10

### Plotting y = sin x

Use your calculator to fill a table from $0^\circ$ to $360^\circ$:

| $x$ | $0^\circ$ | $90^\circ$ | $180^\circ$ | $270^\circ$ | $360^\circ$ |
|---|---|---|---|---|---|
| $\sin x$ | 0 | 1 | 0 | $-1$ | 0 |
| $\cos x$ | 1 | 0 | $-1$ | 0 | 1 |

Plot and join smoothly to get two waves.

*Diagram: Graphs of y equals sin x and y equals cos x from 0 to 360 degrees*
*Sine starts at 0. Cosine starts at 1. Both repeat every 360°.*

### Calculator tip: Table of values
Many calculators have a TABLE mode. Enter $f(x) = \sin x$, the start and end values and the step (such as $\frac{\pi}{6}$, in radian mode), and the calculator fills in the table for you to plot.

### 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 $360^\circ$.
- The centre line of $y = \sin x$ is $y = 0$.

### Key idea: Sine and cosine are the same wave
The cosine graph is the sine graph moved $90^\circ$ to the left. Both waves stay between $-1$ and $1$.

### Watch out: Smooth, not pointy
Sine waves are smooth at the top and bottom. Do not draw them as zigzags.

**Mind map**

```text
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 11: Graphs in radians and the tangent curve
In radians, the period of $\sin x$ and $\cos x$ is $2\pi$. Peaks of $\sin x$ are at $x = \dfrac{\pi}{2}$, $\dfrac{5\pi}{2}$, and so on.

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

**Changing amplitude and period**

For $y = a \sin(bx)$ and $y = a\cos(bx)$:
- amplitude $= |a|$
- period $= \dfrac{2\pi}{b}$

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

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

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

### Year 12: The general wave
$$y = a \sin\big(b(x - c)\big) + d$$

- $|a|$ = amplitude
- $\dfrac{2\pi}{b}$ = 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:

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

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

**Solving with graphs.** The number of solutions of $\sin x = 0.4$ for $0 \leq x \leq 2\pi$ 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 $2\pi$, it is $y = 4\cos x + 3$.

**Solving a model.** For $T = 20 + 6\sin\dfrac{\pi t}{12}$, when is $T$ above 23? Solve $\sin\dfrac{\pi t}{12} = \dfrac{1}{2}$: $\dfrac{\pi t}{12} = \dfrac{\pi}{6}$ or $\dfrac{5\pi}{6}$, 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 $360^\circ$, or $2\pi$.
- Cosine is sine moved left by $90^\circ$.
- $y = \tan x$ has period $\pi$ and asymptotes where $\cos x = 0$.
- $y = a\sin(bx)$: amplitude $|a|$, period $\dfrac{2\pi}{b}$.
- $y = a\sin(b(x - c)) + d$: $c$ shifts right, $d$ moves the centre line.

## Quiz

Try these questions from the chapter quiz.

1. What is the value of $\sin 90^\circ$?
   - A. 0
   - B. 1
   - C. $-1$
   - D. 0.5
   - Answer: B. 1. The sine wave reaches its peak at $90^\circ$.

2. What is the value of $\cos 180^\circ$?
   - A. 0
   - B. 1
   - C. $-1$
   - D. 0.5
   - Answer: C. $-1$. The cosine wave is at its lowest at $180^\circ$.

3. What is the amplitude of $y = \sin x$?
   - A. 1
   - B. 2
   - C. $360^\circ$
   - D. 0
   - Answer: A. 1. Centre line 0, peak 1.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/trigonometric-graphs/#quiz