C. Trigonometric Functions · Sine Functions

Periodic or Sinusoidal? Tables, Graphs, Equations and Restrictions

C3.2About 70 minutes

Learning goals

I can decide whether a table, graph or equation is periodic, and whether it is sinusoidal.
I can test a table for the sine pattern.
I can find the period of a model like h(t) = a sin(kt) + c.
I can state and explain the domain and range that make sense in a real context.

Words to know

restricted domain
The set of inputs that make sense in a real situation, such as 0 ≤ t ≤ 5 min for a 5-minute ride.

discrete
Made of separate values (such as one measurement per month), not every value in an interval.

continuous
Including every value in an interval, with no gaps.

zigzag (triangular) wave
A periodic graph made of straight line segments with sharp corners; periodic but not sinusoidal.

Is it periodic? Is it sinusoidal?

Before you choose a sine model for a situation, you have to decide whether the data or graph is periodic, and if so, whether it is sinusoidal (shaped like the sine curve). A sine model is a poor choice for a pattern with corners or flat parts, or for one that is not periodic at all. You also have to think about what makes sense in the real situation: a ride does not last forever, and a seat cannot be below the ground.

Remember from Lessons 1 and 4

A periodic function repeats at regular intervals. A sinusoidal function is periodic and has the smooth shape of y = sin x, possibly stretched, reflected and translated: y = a sin(x − d) + c.

From a graph

Periodic? Find a piece of the graph that repeats exactly, over and over, at regular intervals.
Sinusoidal? The repeating piece must be a smooth wave: rounded at the top and bottom, symmetric, with the same shape above and below its axis, and no corners or flat sections.

A: periodic and sinusoidal. B: periodic (it repeats) but not sinusoidal (flat parts and sudden jumps). C: not periodic, because the waves keep shrinking. D: periodic and sinusoidal (a sine curve that has been compressed and translated).

From a table of values

With equally spaced inputs, check two things:

Periodic: the outputs repeat in a block (the length of the block in x is the period).
Sinusoidal: the outputs follow the sine pattern. A quick test with steps of one-twelfth of a period (30° steps when the period is 360°): going from the axis toward the max, a sine wave covers half the amplitude in the first step, about 87% after two steps, and all of it after three (because sin 30° = 0.5, sin 60° ≈ 0.866, sin 90° = 1). A straight-line zigzag covers one-third, two-thirds, then all of it.

x	0°	30°	60°	90°	120°	150°	180°	210°	240°	270°	300°	330°	360°
Table P	0	5	8.7	10	8.7	5	0	−5	−8.7	−10	−8.7	−5	0
Table Q	0	3.3	6.7	10	6.7	3.3	0	−3.3	−6.7	−10	−6.7	−3.3	0

Both tables have max 10, min −10 and period 360°. Only one is sinusoidal.

Table P (dark dots on the solid curve) follows y = 10 sin x. Table Q (yellow dots, dashed) climbs in equal steps: it is periodic but its graph is a zigzag, not a sine wave.

Worked example

Example 1: Testing two tables

Use the tables P and Q above. Is each periodic? Is each sinusoidal?

Both start at 0, reach 10 at 90°, 0 at 180°, −10 at 270°, and return to 0 at 360°. If the pattern continues, both are periodic with period 360°, axis y = 0 and amplitude 10.
Table P: after one step (30°) it is at 5 = half the amplitude; after two steps 8.7 ≈ 0.87 × 10. This is the sine pattern: P matches y = 10 sin x. Sinusoidal.
Table Q: it rises 3.3 each step (constant first differences of about 3.3), which is a straight line. Its graph is a zigzag with corners at the max and min. Periodic but not sinusoidal.

P is periodic and sinusoidal (y = 10 sin x); Q is periodic but not sinusoidal.

From an equation

Any equation of the form y = a sin(x − d) + c (with a ≠ 0) is sinusoidal, with period 360°.
In models where the input is time, you will often see equations such as h(t) = 3 sin(30t) + 5. Here 30t is an angle in degrees. One cycle is finished when 30t reaches 360°, at t = 360 ÷ 30 = 12. So the period is 12 time units. These are still sinusoidal. (Courses such as MCR3U, MCT4C and MHF4U study this number in front of t in detail.)
Equations such as y = 2x + 1, y = x², or y = 3(2)ˣ are not periodic: they never repeat.
Adding a non-periodic part to a sine, as in y = sin x + 0.01x, makes the function not periodic: the waves keep drifting upward.

Worked example

Example 2: Classifying equations

Classify each as sinusoidal, periodic but not sinusoidal, or not periodic: (a) y = −4 sin(x − 25°) + 9 (b) y = x² − 3 (c) h(t) = 2 sin(90t) + 6 (d) y = sin x + 0.1x.

(a) It has the form a sin(x − d) + c: sinusoidal, period 360°, amplitude 4, axis y = 9.
(b) A parabola does not repeat: not periodic.
(c) Sinusoidal. The angle 90t completes 360° when t = 4, so the period is 4 time units. Amplitude 2, axis h = 6.
(d) At x = 0° the value is 0; at x = 360° it is sin 360° + 36 = 36; at 720° it is 72. The pattern never returns to the same height, so it is not periodic.

(a) sinusoidal (b) not periodic (c) sinusoidal, period 4 (d) not periodic.

Restrictions from the context

The function y = a sin(x − d) + c has domain {x ∈ ℝ}. A real situation usually restricts it:

Domain: time starts at 0 and ends when the activity ends. A Ferris wheel ride of 3 full turns has domain 0° ≤ θ ≤ 1080° (in degrees turned), or 0 ≤ t ≤ (ride time) in seconds.
Range: the model's range is from min to max. In context, check that the values make sense, for example a height cannot be below ground and a number of hours of daylight must be between 0 and 24.
Discrete or continuous: if values are measured only once a day or once a month (like daylight hours on the 21st), the domain is a set of separate values (whole numbers of days or months), even if we draw a smooth curve through them.

A three-turn Ferris wheel ride, h = 10 sin(θ − 90°) + 12. The domain is restricted to 0° ≤ θ ≤ 1080° (the ride starts and ends at the bottom). The dots mark the start and end of the ride; the shaded band shows the range, 2 ≤ h ≤ 22.

Worked example

Example 3: Domain and range in context

A Ferris wheel seat's height is h(θ) = 10 sin(θ − 90°) + 12, where θ is the angle turned. A ride is exactly 3 full turns. State the domain and range in context.

Each turn is 360°, so 3 turns = 1080°. Domain: {θ ∈ ℝ | 0° ≤ θ ≤ 1080°}. (Before boarding and after getting off, the model does not apply.)
Min = 12 − 10 = 2 m, max = 12 + 10 = 22 m. During 3 full turns the seat reaches both. Range: {h ∈ ℝ | 2 ≤ h ≤ 22}.
Check: all heights are above ground (2 m > 0), so the model makes sense.

Domain 0° ≤ θ ≤ 1080°; range 2 ≤ h ≤ 22 (metres).

Just for fun

Riddle: I repeat myself every 24 hours, I'm smooth-ish but have flat parts at night, and I'm brightest at noon. What am I?

The graph of sunlight on your window: periodic, but not quite sinusoidal.

Hands-on activity

Periodic or sinusoidal? A kitchen investigation

You need: a glass of water, a spoon, a phone or clock with a second hand, paper and a pencil.

Second hand height. Imagine a point at the tip of a clock's second hand. Sketch its height above the centre of the clock for 2 min. Is it periodic? Sinusoidal? What are its period and domain if you watch for exactly 2 min?
Stirring. Stir the water steadily in circles and watch one bubble or bit of foam. Sketch its distance from your side of the glass against time for a few turns. Is it periodic? Sinusoidal? What happens after you stop stirring?
Dripping tap (just imagine it, or use a bottle that drips slowly into the sink). Sketch the water height in a cup that fills steadily and is emptied every 30 s. Is it periodic? Sinusoidal?

What to look for: (1) periodic and sinusoidal, period 60 s, domain 0 ≤ t ≤ 120 s, range from −(hand length) to +(hand length). (2) roughly periodic and roughly sinusoidal while stirring, but the waves die down after you stop, so it is no longer periodic. (3) periodic (period 30 s) but not sinusoidal: a sawtooth that rises in a straight line and drops suddenly.

Your turn

Is y = 5 sin(x + 40°) − 2 sinusoidal? Give its period, amplitude and axis.

A table with x in steps of 30° has values 4, 6, 7.5, 8, 7.5, 6, 4, 2, 0.5, 0, 0.5, 2, 4. Is it sinusoidal? Explain.

Is the height of a bouncing ball (that bounces lower each time) periodic?

h(t) = 6 sin(45t) + 10. Find the period.

A Ferris wheel ride lasts 4 full turns. State the domain in degrees turned.

Daylight hours in a town are recorded on the 1st of every month for a year. Is the domain continuous or discrete? Explain.

More practice: warm-up

Is y = 3x + 2 periodic?

Is y = −sin x sinusoidal?

A graph repeats but has sharp corners. Periodic? Sinusoidal?

Can the domain of a Ferris wheel model include negative time?

h(t) = 2 sin(30t) + 3. Period?

More practice: on your own

A wheel has h = 8 sin(θ − 90°) + 10, and a ride lasts 2.5 turns. State the domain and range in context.

A model for the depth of water at a dock is d(t) = 1.5 sin(30t) + 1. What does the model predict for the lowest depth, and why is this a problem?

x: 0, 1, 2, 3, 4, 5, 6, 7, 8; y: 3, 7, 3, −1, 3, 7, 3, −1, 3. Periodic? Sinusoidal? Explain using the pattern of values.

Classify y = 2 sin x + x.

A swimmer's distance from the wall is recorded for a 6-minute swim. Give a sensible domain, and explain whether the graph is sinusoidal.

Challenge

For h(t) = 5 sin(20t) + 7, with t in seconds, find the period and decide whether h is ever negative.

Explain why a table that only shows the values at the axis, max and min (every quarter-period) cannot prove a function is sinusoidal.

A Ferris wheel ride starts with the seat at the bottom, and the model is h = 12 sin(θ − 90°) + 14. A rider gets off after the seat has turned 990°. State the domain and the range of the model for this ride.

Check your understanding

Quiz

Which graph is periodic but not sinusoidal?

Which equation is not periodic?

h(t) = 4 sin(60t) + 9. What is the period?

A Ferris wheel ride is 2 full turns. What is the domain in degrees turned?

In a table with steps of 30° (period 360°, axis 0, amplitude 10), which values after 0 are sinusoidal?

A tide model gives a minimum depth of −0.3 m. What does this tell you?

For parents and teachers

Big idea: Identifying periodic and sinusoidal functions from tables, graphs and equations, and explaining the restrictions a real context puts on the domain and range (C3.2).

How to help: Show your teen graphs from the news or a weather app (temperature over a week, tides, electricity use) and ask: "Periodic? Sinusoidal? What domain makes sense?"

Common difficulties:

Calling every periodic graph sinusoidal. Look for corners, flat parts or straight segments.
Forgetting that time cannot be negative and a ride has an end.
Accepting a model's prediction (like a negative depth) without checking whether it makes sense.