C. Trigonometric Functions · Sine Functions

Key Properties of f(x) = sin x

C2.4About 65 minutes

Learning goals

I can sketch f(x) = sin x quickly using five key points.
I can state the period, amplitude, axis, maximum and minimum of f(x) = sin x.
I can state its domain, range and intercepts.
I can state where it is increasing and decreasing.

Words to know

zero
An x-value where the function's output is 0 (an x-intercept).

increasing interval
An interval of x on which the graph rises from left to right.

decreasing interval
An interval of x on which the graph falls from left to right.

five key points
The points (0°, 0), (90°, 1), (180°, 0), (270°, −1) and (360°, 0) that outline one cycle of y = sin x.

set notation
A way to write a domain or range, such as {y ∈ ℝ | −1 ≤ y ≤ 1}.

Getting to know the sine curve

When you meet a new function, you describe it with the same list of features every time: its domain and range, intercepts, maximum and minimum, and where it rises and falls. For a periodic function you add its cycle, period, amplitude and axis. Today you will build that list for f(x) = sin x and learn to sketch the graph quickly by hand. Every transformed sine graph in this unit starts from this one.

Remember from the Quadratic Functions unit

The domain is the set of all input values; the range is the set of all output values. We write them in set notation, for example {x ∈ ℝ} (all real numbers) or {y ∈ ℝ | y ≥ 2}. A zero (x-intercept) is an input where the output is 0.

One cycle of f(x) = sin x with its five key points. The curve rises on the green-shaded intervals and falls on the yellow one.

Property	f(x) = sin x
period	360°
amplitude	1
equation of the axis	y = 0
maximum value	1, at x = 90° (and 90° + 360°n)
minimum value	−1, at x = 270° (and 270° + 360°n)
x-intercepts (zeros)	0°, 180°, 360°, … (every multiple of 180°)
y-intercept	0
domain	{x ∈ ℝ}
range	{y ∈ ℝ \| −1 ≤ y ≤ 1}
increasing (0° to 360°)	0° < x < 90° and 270° < x < 360°
decreasing (0° to 360°)	90° < x < 270°

Key properties of f(x) = sin x (angles in degrees; n is any integer).

Reading the properties

Cycle and period. One cycle runs from 0° to 360°: up to the maximum, down through the axis to the minimum, and back up. Period = 360°.
Maximum, minimum, axis, amplitude. Max 1, min −1, so the axis is y = (1 + (−1)) ÷ 2 = 0 (the x-axis) and the amplitude is (1 − (−1)) ÷ 2 = 1.
Intercepts. The graph crosses the x-axis wherever sin x = 0: at 0°, 180°, 360°, and every multiple of 180°, including negative ones. The y-intercept is sin 0° = 0.
Domain. You can take the sine of any angle, including negative angles and angles larger than 360°, so the domain is all real numbers.
Range. sin x is never above 1 or below −1, and it takes every value in between, so the range is −1 ≤ y ≤ 1.
Increasing and decreasing. Reading left to right, the curve goes up from 0° to 90°, down from 90° to 270°, and up again from 270° to 360°. These are written as intervals of x, not y.

Two cycles of f(x) = sin x, from −360° to 360°. Yellow dots: zeros every 180°. Blue dots: maximums at −270° and 90°, minimums at −90° and 270°, each 360° apart.

Worked example

Example 1: Listing properties over a larger interval

For f(x) = sin x with −360° ≤ x ≤ 360°, list (a) the zeros, (b) the x-values of the maximums and minimums, (c) the intervals where f is increasing.

(a) Zeros are every 180°: −360°, −180°, 0°, 180°, 360°.
(b) Maximums are at 90° plus or minus multiples of 360°: −270° and 90°. Minimums are at 270° plus or minus 360°: −90° and 270°.
(c) The curve rises from each minimum to the next maximum. At −360° the graph looks exactly like it does at 0° (360° earlier), so it starts by rising to the maximum at −270°. It increases from −360° to −270°, from the minimum at −90° to the maximum at 90° (−90° to 90°), and from the minimum at 270° to the end, 270° to 360°.

Zeros −360°, −180°, 0°, 180°, 360°; maximums at −270° and 90°; minimums at −90° and 270°; increasing on −360° < x < −270°, −90° < x < 90° and 270° < x < 360°.

Sketching sin x by hand: the five key points

You do not need a table of 17 values to sketch the graph. Use the five key points of one cycle:

(0°, 0), (90°, 1), (180°, 0), (270°, −1), (360°, 0)

They divide the cycle into four equal quarters of 90°. Join them with a smooth, rounded curve (not straight lines), and repeat the shape for more cycles.

Three steps for a quick sketch: mark the quarter points, plot the five key points, then join them with a smooth wave.

Worked example

Example 2: Using properties to answer questions

Without a calculator, use the properties of f(x) = sin x to answer: (a) Is sin 135° greater than or less than sin 45°? (b) Which is larger, sin 100° or sin 170°? (c) What is the maximum value of 3 + sin x?

(a) 135° = 180° − 45°, and sin(180° − θ) = sin θ, so they are equal.
(b) From 90° to 270° the function is decreasing, so a larger angle in that interval gives a smaller sine. 100° < 170°, so sin 100° is larger.
(c) The largest value of sin x is 1, so the largest value of 3 + sin x is 3 + 1 = 4.

(a) Equal (b) sin 100° (c) 4

Just for fun

Why doesn't the sine function ever get a swelled head?

No matter how well things go, its range keeps it between −1 and 1.

Hands-on activity

Human sine wave

You need: a hallway or a room with space, masking tape (or a long string), five sticky notes and a pencil. A grown-up or sibling helps.

Lay a line of tape (the x-axis) about 3 m long along the floor. Put sticky notes at equal spaces labelled 0°, 90°, 180°, 270° and 360°.
Walk slowly along the tape, holding one hand out in front of you. Use your hand height to show sin x: at 0° hold it at waist height (the axis), at 90° raise it as high as you can, at 180° back to waist height, at 270° as low as you can, at 360° back to waist height.
Your helper calls out an angle such as 45° or 300° while you walk, and you hold your hand at the right height. Say whether the value is increasing or decreasing.

What to look for: your hand moves smoothly, not in jerks; it is rising between 0° and 90° and between 270° and 360°, and falling between 90° and 270°; at 45° it is about 70% of the way up and at 300° about 87% of the way down.

Your turn

State the period, amplitude and equation of the axis of f(x) = sin x.

State the domain and range of f(x) = sin x in set notation.

List the zeros of f(x) = sin x from 0° to 720°.

On 0° ≤ x ≤ 360°, where is sin x decreasing?

Find the x-values of all maximums of f(x) = sin x from −720° to 720°.

Sketch one cycle of f(x) = sin x from 0° to 360° using the five key points. Which points did you use?

More practice: warm-up

What is the y-intercept of y = sin x?

What is the maximum value of sin x?

At which x from 0° to 360° does sin x reach its minimum?

How far apart are neighbouring zeros of y = sin x?

Is sin x increasing or decreasing at x = 300°?

More practice: on your own

For −360° ≤ x ≤ 0°, state where f(x) = sin x is decreasing.

Without a calculator, order from least to greatest: sin 10°, sin 80°, sin 200°, sin 270°.

What are the maximum and minimum values of y = 5 − sin x? At which x from 0° to 360° does each occur?

How many times does y = sin x cross the line y = 0.5 for 0° ≤ x ≤ 1080°?

Explain why the range of y = sin x is not {y ∈ ℝ}.

Challenge

Find the zeros, maximum points and minimum points of y = sin x for −540° ≤ x ≤ −180°.

A function g has the same shape as sin x but its maximum is 7 and its minimum is 3. State its amplitude, axis and range.

Is the statement 'sin x is increasing on 0° < x < 180°' true? Explain using the graph.

Check your understanding

Quiz

What is the range of f(x) = sin x?

Which list gives the zeros of y = sin x from 0° to 360°?

On 0° ≤ x ≤ 360°, sin x is increasing on

Which is not one of the five key points of one cycle of y = sin x?

What is the domain of f(x) = sin x?

Without a calculator, which is greater: sin 110° or sin 160°?

For parents and teachers

Big idea: Sketching f(x) = sin x and describing its key properties: cycle, domain, range, intercepts, amplitude, period, maximum and minimum, and increasing/decreasing intervals (C2.4).

How to help: Ask your teen to sketch y = sin x from memory using the five key points, then quiz them: "Where is it zero? Where is it highest? Where is it rising?"

Common difficulties:

Writing increasing/decreasing intervals with y-values. They are always intervals of x.
Saying the period is 180° (the distance between zeros) instead of 360°.
Drawing a zigzag with sharp corners instead of a smooth, rounded curve.