D. Trigonometric Functions · Sinusoidal FunctionsCollecting Sinusoidal Data and Finding a Model
D3.1 D3.2About 75 minutes
Learning goals- I can collect sinusoidal data from an experiment or a secondary source and graph it.
- I can decide whether data are periodic, sinusoidal, or neither.
- I can find a sinusoidal model for data and check how well it fits.
- I can state a sensible domain and range for a model in its real-world context.
Words to know
secondary source
Data that someone else collected and published, such as climate normals or tide tables.
model
An equation that describes data closely enough to explain or predict them.
curve of best fit
A smooth curve that follows the overall pattern of data points, with points scattered close to it on both sides.
sinusoidal regression
A technology feature that finds the sinusoidal equation that fits data best.
frequency
The number of cycles per unit of time. A frequency of 60 Hz (hertz) means 60 cycles per second; period = 1 ÷ frequency.
From real measurements to an equation
In the last lesson you wrote equations for perfect graphs. Real data are never perfect: measurements have small errors and nature does not repeat itself exactly. Today you will collect data (or find them), graph them, and build a sinusoidal model: an equation that fits the data well enough to describe and predict them.
Where sinusoidal data come from
Experiments you can do: a pendulum, a weight bouncing on a spring, a mark on a turning wheel, or a motion sensor tracking someone walking in a circle. A phone app that shows sound waves from the microphone can display the nearly sinusoidal wave of a tuning fork or a hummed note.
Secondary sources: monthly average temperatures (Environment and Climate Change Canada publishes climate normals for many communities), predicted tide heights (published by the Canadian Hydrographic Service of Fisheries and Oceans Canada), and sunrise and sunset times (the National Research Council Canada has a sunrise/sunset calculator).
Electricity: the outlets in Canadian homes supply alternating current. The voltage rises and falls sinusoidally, going through 60 cycles every second (60 Hz). (Never experiment with a wall outlet: household current is dangerous.)
Is it sinusoidal?
Before fitting a sine curve, check that the data are:
- periodic: the pattern repeats at regular intervals;
- smooth: no corners, jumps or flat stretches;
- symmetric: the maxima are about as far above the axis as the minima are below it, and the graph rises and falls in about the same time.
If the data repeat but fail 2 or 3 (a heartbeat signal, the number of people in a cafeteria each hour), they are periodic but not sinusoidal, and a sine model would be poor.
Building a model from data
- Graph the data and check that a sinusoidal model makes sense.
- Estimate the max and min. If there are several cycles, average the maxima and average the minima.
- Find c = (max + min) ÷ 2 and |a| = (max − min) ÷ 2.
- Find the period (average several cycles if you can) and k = 360 ÷ period.
- Choose sine or cosine and find d from a maximum, a minimum or an axis crossing.
- Check the fit: compare model values with the data. The differences should be small and should not all be on one side.
Many graphing tools can also do sinusoidal regression, which finds the best-fitting equation automatically. Your hand-made model should be close to it.
| t (s) | 0 | 0.25 | 0.5 | 0.75 | 1.0 | 1.25 | 1.5 | 1.75 | 2.0 |
|---|
| d (cm) | 12.0 | 8.4 | 0.2 | −8.6 | −11.9 | −8.5 | −0.1 | 8.4 | 11.8 |
Worked exampleExample 1: Modelling a pendulum
Use the table to write a model for the pendulum's position d (cm) after t seconds, and check it at t = 0.25 s.
- The data rise and fall smoothly and symmetrically, so a sinusoidal model makes sense.
- Max ≈ 12 (average of 12.0 and 11.8 is 11.9, so about 12); min ≈ −12. So c = 0 and a = 12.
- The pendulum is back at its start after 2.0 s: period = 2 s, so k = 360 ÷ 2 = 180 (degrees per second).
- It starts at a maximum at t = 0, so use cosine with d = 0: d = 12 cos(180t).
- Check t = 0.25: 12 cos(45°) ≈ 8.5 cm. The measurement was 8.4 cm: a difference of only 0.1 cm.
The pendulum data (circles) and the model d = 12 cos(180t) (curve), extended to 4 s. The data fall almost exactly on the curve.| Month | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec |
|---|
| Data T (°C) | −6 | −5 | 0 | 7 | 13 | 19 | 22 | 21 | 17 | 10 | 4 | −2 |
| Model T (°C) | −6 | −4.1 | 1 | 8 | 15 | 20.1 | 22 | 20.1 | 15 | 8 | 1 | −4.1 |
Worked exampleExample 2: Modelling monthly temperatures
Write a model for the sample temperature data, compare it with the data, and state a sensible domain and range.
- Max 22 °C (July, m = 7); min −6 °C (January, m = 1).
- c = (22 + (−6)) ÷ 2 = 8; a = (22 − (−6)) ÷ 2 = 14.
- The cycle is one year: period 12 months, so k = 360 ÷ 12 = 30.
- A clear maximum in July: cosine with d = 7. T = 14 cos(30(m − 7)) + 8.
- Compare (see the table): the model is within about 3 °C every month. It is a bit too warm in spring and a bit too cool in fall, because real temperatures stay warm a little longer after July. Overall it is a good model.
- Domain: the model is for months of a year, 1 ≤ m ≤ 12 (or any m if we repeat the yearly pattern). Range of the model: −6 ≤ T ≤ 22.
T = 14 cos(30(m − 7)) + 8, a good fit within about 3 °C.
Sample monthly temperature data (circles) with the model T = 14 cos(30(m − 7)) + 8. The data stay close to the curve on both sides.Worked exampleExample 3: The voltage in a wall outlet
In Canada, household voltage is described as 120 V at 60 Hz. The voltage actually swings between about +170 V and −170 V (the "120 V" is a kind of average). Write a model for the voltage V after t milliseconds, starting at 0 V and increasing, and find V at t = 5 ms.
- 60 cycles per second means the period is 1/60 s = 1000/60 ms ≈ 16.7 ms.
- k = 360 ÷ (1000/60) = 360 × 60 ÷ 1000 = 21.6 degrees per millisecond.
- a = 170, c = 0, and it starts at 0 going up, so use sine with d = 0: V = 170 sin(21.6t).
- At t = 5 ms: 21.6 × 5 = 108°, and 170 sin 108° ≈ 170(0.951) ≈ 162 V.
V = 170 sin(21.6t); about 162 V at t = 5 ms.
Just over two cycles of the model V = 170 sin(21.6t) for a Canadian household outlet. One full cycle takes only about 16.7 thousandths of a second.Domain and range in context
The equation of a sinusoidal function has domain {x ∈ ℝ}, but a model only makes sense where the situation does. Time usually starts at 0. A pendulum slowly loses height because of friction, so a model with constant amplitude is only good for a short time. A temperature model gives typical values, not the weather on a particular day. Always state the domain and range that fit the situation.
Just for funThe voltage in a Canadian wall outlet runs through 60 complete cycles every second. In one second, about the time it takes to say "sinusoidal" slowly, it goes up and down 60 times.
Hands-on activityDaylight data from a secondary source
You need: internet access, graph paper (or a spreadsheet), a pencil and a calculator.
- Use a sunrise and sunset calculator (for example, the one from the National Research Council Canada) or a weather app to find the sunrise and sunset times for your community on the 21st of each month this year.
- For each date, compute the hours of daylight: sunset − sunrise. Convert minutes to decimal hours (for example, 9 h 30 min = 9.5 h).
- Graph daylight (vertical) against the day number of the year (Jan 21 = day 21, Feb 21 = day 52, Mar 21 = day 80, and so on).
- Decide whether the data look sinusoidal. Find the max, min, c, |a|, the period and d, and write a model.
- Check your model for two months. How close is it?
- Optional: compare with a friend or relative in another part of Canada. Whose graph has the bigger amplitude?
What to look for: a smooth yearly wave with period about 365 days, a maximum near June 21 (day 172) and an axis of about 12 h. In southern Ontario the amplitude is a little over 3 h. Communities farther north have a larger amplitude: longer summer days and shorter winter days.
Your turn
1Data: x = 0, 1, 2, 3, 4, 5, 6, 7, 8 gives y = 4, 7, 4, 1, 4, 7, 4, 1, 4. Write a sine model.
Max 7, min 1: a = 3, c = 4. Period 4, so k = 90. At x = 0 it is on the axis going up: y = 3 sin(90x) + 4. Check x = 1: 3 sin 90° + 4 = 7 ✓.
2A weight bouncing on a spring is 40 cm above the floor at t = 0 (its highest point), 30 cm at t = 0.4 s (its lowest point), and 40 cm again at t = 0.8 s. Write a model for its height h after t seconds.
c = 35, a = 5. Period 0.8 s, so k = 360 ÷ 0.8 = 450. Starts at a max: h = 5 cos(450t) + 35.
3Canadian household current has a frequency of 60 Hz. What is its period in milliseconds, and what is k (in degrees per millisecond)?
Period = 1000 ÷ 60 ≈ 16.7 ms; k = 360 ÷ (1000/60) = 21.6.
4Which are best described as sinusoidal, and which as periodic but not sinusoidal? (a) the sound wave from a tuning fork (b) the electrical signal of a heartbeat (c) the number of people in a school cafeteria each hour over a week
(a) Sinusoidal (a smooth, regular wave). (b) Periodic but not sinusoidal (sharp spikes). (c) Periodic but not sinusoidal (sudden jumps at lunch times, flat at night).
5A pendulum is modelled by d = 12 cos(180t), and it is recorded for 10 s. State a sensible domain and range.
Domain 0 ≤ t ≤ 10; range −12 ≤ d ≤ 12 (in cm).
6Use T = 14 cos(30(m − 7)) + 8 to predict the average temperature in April (m = 4). Compare it with the sample data.
30(4 − 7) = −90°, cos(−90°) = 0, so T = 8 °C. The data value is 7 °C, so the model is 1 °C too warm.
7Why is it a good idea to average several maxima and several minima when building a model from real data?
Real measurements vary a little from cycle to cycle and contain small errors. Averaging reduces the effect of any one unusual value, so a and c are more reliable.
More practice: warm-up
1Data have a max of 18 and a min of 6. Find a (a > 0) and c.
2The period of a pendulum is 0.5 s. Find k.
3Name one Canadian secondary source of sinusoidal data.
Sample answers: climate normals from Environment and Climate Change Canada; tide tables from the Canadian Hydrographic Service; the NRC sunrise/sunset calculator.
4A frequency of 60 Hz means how many cycles per second? What is the period in seconds?
60 cycles per second; period 1/60 s.
5A graph repeats regularly but has flat tops and straight sides. Is a sinusoidal model a good choice?
No. It is periodic but not smooth and rounded like a sine wave.
More practice: on your own
1Sample tide data: high tide 10.5 m at 2:00, low tide 1.5 m at 8:12, high tide 10.5 m at 14:24. Write a model for the height h (m) t hours after midnight.
c = 6, a = 4.5. Period = 14:24 − 2:00 = 12.4 h, so k = 360 ÷ 12.4 ≈ 29.0. Max at t = 2: h = 4.5 cos(29.0(t − 2)) + 6.
2Use this data (period 40 s) to write a model: t = 0, 5, 10, 15, 20, 25, 30, 35, 40 gives h = 3, 6.5, 15, 23.5, 27, 23.5, 15, 6.5, 3.
c = 15, |a| = 12, k = 360 ÷ 40 = 9. It starts at a minimum, so use a reflected cosine: h = −12 cos(9t) + 15. Check t = 5: −12 cos 45° + 15 ≈ 6.5 ✓.
3The valve on a bicycle wheel is 30 cm from the centre of the wheel, and the centre is 33 cm above the ground. The wheel turns once every 0.5 s, and the valve is at its lowest point at t = 0. Write a model for the valve's height, and state its range.
c = 33, |a| = 30, k = 360 ÷ 0.5 = 720. Starts at a minimum: h = −30 cos(720t) + 33. Range 3 ≤ h ≤ 63 (cm). (This assumes the bike moves at a steady speed.)
4A tuning fork vibrates 256 times per second. Find the period of its sound wave in milliseconds and k in degrees per second.
Period = 1 ÷ 256 s ≈ 0.0039 s ≈ 3.9 ms. k = 360 × 256 = 92 160 degrees per second.
5In Example 2, list the months where the model is too warm and the months where it is too cool. What pattern do you notice?
Too warm (model above data): February to June (for example, May: 15 vs 13). Too cool (model below data): August to December (for example, November: 1 vs 4). Exactly right: January and July. Pattern: real temperatures lag behind the model: spring warms up later and fall stays warm longer, so the data's warm season is shifted a little later than the model's.
Challenge
1Many countries in Europe use household electricity described as 230 V at 50 Hz. The peak voltage is 230√2 V. Write a model for the voltage after t milliseconds, starting at 0 V and increasing.
Peak: 230√2 ≈ 325 V. Period = 1000 ÷ 50 = 20 ms, so k = 360 ÷ 20 = 18. V = 325 sin(18t).
2A motion sensor tracks a swinging pendulum. Its distance from the sensor is a minimum of 0.8 m at t = 0.3 s and then a maximum of 1.4 m at t = 1.3 s. Write a model for the distance D (m).
c = 1.1, |a| = 0.3. Min to max is half a period: 1.0 s, so the period is 2 s and k = 180. Start at the minimum: D = −0.3 cos(180(t − 0.3)) + 1.1.
3A pendulum's swings slowly get smaller. Explain why a model like d = 12 cos(180t) works well for the first few seconds but not for several minutes. What would a better model need?
Friction and air resistance take energy away, so the amplitude shrinks over time, while the model's amplitude stays 12 cm forever. After several minutes the real swings are much smaller than 12 cm. A better model would need an amplitude that decreases with time (for example, a sine function multiplied by a function that gets smaller over time; Grade 12 courses explore combinations like this).
Quiz
1. Which of these is best modelled by a sinusoidal function?
2. A pendulum has a period of 2 s. In a model d = a cos(kt), what is k?
3. Household current in Canada has a frequency of 60 Hz. About how long is one cycle?
4. Data have a maximum of 15 and a minimum of 3. What is the equation of the axis of a sinusoidal model?
5. A pendulum is recorded from t = 0 to t = 30 s. What is a sensible domain for its model?
6. Which data are periodic but not sinusoidal?
Big idea: D3.1 asks students to collect data that can be modelled as a sinusoidal function, from experiments or secondary sources, and to graph them. D3.2 asks them to identify sinusoidal and periodic functions in real situations and to explain domain and range restrictions.
How to help: The daylight activity works well as a shared project: look up your community's sunrise and sunset times together. Ask your teen to explain each number in their model ("Why is c about 12?").
Safety: Do not let anyone experiment with wall outlets or open electrical devices. The voltage example is for calculation only.
Common difficulties:
- Using a single unusual data point as the max or min. Encourage averaging.
- Forgetting units for k (degrees per second, per hour, per month) when time is the variable.
- Expecting a perfect fit. Real data scatter around a good model; the differences should be small and on both sides.
- Giving {x ∈ ℝ} as the domain of a real-world model. Ask, "When does this situation start and end?"