C. Trigonometric Functions · Sine Functions

Collecting and Graphing Sinusoidal Data

C3.1 C3.2About 75 minutes

Learning goals

I can plan and carry out an investigation that collects periodic data.
I can use primary and secondary sources of sinusoidal data.
I can graph data, draw a curve of best fit and decide whether it is sinusoidal.
I can write a model y = a sin(x − d) + c from data and test it.

Words to know

primary data
Data you collect yourself by measuring or observing.

secondary data
Data collected by someone else, such as published tables or databases.

motion sensor
A device that measures the distance to an object many times per second and sends it to a calculator or computer.

curve of best fit
A smooth curve that passes as close as possible to most of the data points.

sinusoidal regression
A technology tool that calculates the sine curve that best fits a set of data.

Becoming a data collector

Up to now, the periodic data in this unit came from tables in the lessons. Today you collect data that can be modelled by a sine function, organize it, graph it, and fit a model. Scientists and engineers do exactly this when they measure tides, sound waves, alternating current in the electrical outlets of your home, or the motion of a bridge in the wind.

Remember from Grade 10

When you collected data for a quadratic relation, you made a table, plotted a scatter plot, and drew a curve of best fit that passed close to most points without having to touch every one. You will do the same thing with a wave-shaped curve.

Primary and secondary sources

Primary data are measurements you make yourself: with a ruler or measuring tape, a stopwatch, a phone video, or a motion sensor connected to a graphing calculator or computer (a motion sensor measures the distance to an object many times per second).
Secondary data come from someone else's measurements: tide tables from Fisheries and Oceans Canada, hours of daylight from a sunrise and sunset table, or data from Statistics Canada.

Planning an investigation

Decide the independent variable (angle turned, or time) and the dependent variable (height, distance, depth).
Choose equal intervals for the independent variable, small enough to show the shape (at least 8 to 12 readings per cycle).
Collect at least one full cycle, two if you can.
Record the data in a table with units, then make a scatter plot.
Decide if the pattern looks sinusoidal. If it does, sketch a smooth curve of best fit and read its max, min, axis, amplitude and period.
Write a model and check it against a few data points.

A bicycle wheel investigation (the bike is upside down and held steady). The height of a tape mark on the tire is measured each time the wheel turns another 30°.

θ (°)	0	30	60	90	120	150	180	210	240	270	300	330	360
h (cm)	36	52	65	69	65	53	36	20	8	3	7	19	36

Invented sample data for practice (not real measurements): height of the tape mark above the floor, in centimetres.

Worked example

Example 1: Modelling the wheel data

Use the invented sample wheel data in the table above (made up for practice, not real measurements). (a) Is the data sinusoidal? (b) Find the max, min, axis and amplitude. (c) Write a model h = a sin(θ − d) + c. (d) Check the model at θ = 60° and θ = 240°.

(a) The heights rise from 36 to a maximum, fall to a minimum and return to 36 after 360°. From the axis, the first step rises 16 (about half of 33) and the second 29 (about 0.87 × 33). The pattern is smooth and symmetric: sinusoidal, period 360°.
(b) Max = 69 cm (at 90°), min = 3 cm (at 270°). Axis: h = (69 + 3) ÷ 2 = 36. Amplitude = (69 − 3) ÷ 2 = 33 cm (the radius of the wheel).
(c) At θ = 0° the mark is on the axis and rising, which is exactly how y = sin θ starts, so d = 0°. a = 33 and c = 36: h = 33 sin θ + 36.
(d) θ = 60°: 33(0.866) + 36 ≈ 64.6 cm; measured 65 cm ✓. θ = 240°: 33(−0.866) + 36 ≈ 7.4 cm; measured 8 cm ✓. In real measurements, small differences like these come from measurement error.

Sinusoidal; max 69, min 3, axis h = 36, amplitude 33; model h = 33 sin θ + 36.

The invented sample data points (dots) and the model h = 33 sin θ + 36 (dashed). The model passes very close to every point.

Data that depend on time

Many sinusoidal data sets use time as the independent variable, so the period is not 360. A useful trick: convert time to degrees by matching one full period to 360°.

For yearly data recorded once a month, one month = 360° ÷ 12 = 30°. For a tide with period 12 h, one hour = 360° ÷ 12 = 30°. Then you can model with the a, d and c you already know.

Worked example

Example 2: Modelling daylight (secondary data)

Lesson 2 gave the approximate hours of daylight in Ottawa on the 21st of each month: 9.3, 10.7, 12.1, 13.8, 15.1, 15.7, 15.2, 13.9, 12.3, 10.7, 9.3, 8.7 (January to December). Let m be the month number (January = 1) and θ = 30m degrees. Write a model D = a sin(θ − d) + c and test it for April.

Max 15.7 (June, m = 6), min 8.7 (December, m = 12). c = (15.7 + 8.7) ÷ 2 = 12.2, a = (15.7 − 8.7) ÷ 2 = 3.5.
The maximum is at m = 6, so θ = 30 × 6 = 180°. For y = sin θ the maximum is at 90°, so d = 180° − 90° = 90°.
Model: D = 3.5 sin(30m − 90°) + 12.2 (here 30m is the angle in degrees).
Test April (m = 4): D = 3.5 sin(120° − 90°) + 12.2 = 3.5 sin 30° + 12.2 = 1.75 + 12.2 ≈ 14.0 h. The table gives 13.8 h. Close, so the model fits well.

D = 3.5 sin(30m − 90°) + 12.2; it predicts about 14.0 h for April (actual about 13.8 h).

Ottawa daylight data (dots) with the model D = 3.5 sin(30m − 90°) + 12.2 (dashed). Real daylight is not a perfect sine wave, but the model is within about 0.25 h (15 min) of the table for every month.

Using technology

A graphing calculator or spreadsheet can plot the data, and many can calculate a sinusoidal regression (a sine curve of best fit). When you use one, set the angle mode to degrees and compare the result with your own estimates of a, c and d. With a motion sensor, you can collect a pendulum's distance from the sensor many times per second and see a sine-shaped graph appear live.

Just for fun

Measuring the same wheel twelve times might seem dull.

But it does have its ups and downs.

Hands-on activity

Collect your own sinusoidal data

Choose one investigation. A grown-up should help with the bicycle (keep fingers away from spokes and chain).

A. Bicycle wheel or lazy Susan. Put a piece of tape on the tire (or on the edge of a lazy Susan, measuring its distance from the table's front edge instead of height). Turn 30° at a time for a full turn (use a protractor template or count spokes) and measure. Follow Example 1.

B. Pendulum. Tie a small weight (a washer or a few keys) to a 1 m string and hang it from a doorway. Pull it 20 cm to one side and release. Film it with a phone next to a metre stick on the floor. Play back frame by frame and record the horizontal position every 0.2 s for 2 swings. (It slowly loses height, so use only the first few swings.)

C. Daylight (secondary data). Look up sunrise and sunset times for your town on the 21st of each month of last year (a weather service or almanac). Compute daylight hours and follow Example 2.

For each: make a table, a scatter plot, a curve of best fit, and a model. Test the model at two points.

What to look for: A gives amplitude equal to the radius and axis at the hub height. B gives a period of about 2 s for a 1 m pendulum and an amplitude close to 20 cm. C gives a period of 12 months and an amplitude of a few hours, larger the farther north you live.

Your turn

Is the daylight table in Example 2 primary or secondary data? Is the wheel data?

A data set for a turning wheel has max 92 cm and min 10 cm. Find a and c for h = a sin(θ − d) + c.

For that wheel, the tape starts at the bottom (θ = 0°). Find d and write the model.

Why should you collect at least 8 readings per cycle?

Monthly data has period 12 months. How many degrees is one month in the θ = 30m trick? Why?

Using D = 3.5 sin(30m − 90°) + 12.2, predict Ottawa's daylight on September 21 (m = 9).

More practice: warm-up

Name one tool for collecting primary periodic data.

Name one source of secondary periodic data.

What is the dependent variable in the wheel investigation?

Max 20, min 4. Amplitude?

Max 20, min 4. Axis?

More practice: on your own

A Ferris wheel data set (height vs angle) has max 42 m at 180° and min 2 m at 0°. Write a model and test it at 90°.

A student's wheel data: θ = 0°, 90°, 180°, 270°, 360° gives h = 30, 55, 30, 5, 30. A second student measured every 30°. Why are the second student's data better?

Tide data at a wharf: high tide 4.6 m, low tide 0.8 m, period 12 h, high tide at t = 3 h. Using θ = 30t degrees, write a model.

Using the tide model above, predict the depth at t = 7 h.

In the pendulum investigation, why should you use only the first few swings?

Challenge

Write the Ottawa daylight model with a negative a instead, and check it at m = 12.

A data set has max 15 at x = 50° and the next min at x = 230°, with min 3. Write a model y = a sin(x − d) + c (a > 0) and give the value at x = 140°.

For the Ottawa model, explain one reason the model and the table differ by up to about 0.25 h in some months.

Check your understanding

Quiz

Which is an example of secondary data?

Wheel data have max 50 cm and min 6 cm. What is the amplitude?

A tape mark starts at the bottom of a wheel (θ = 0°). Which d should the model h = a sin(θ − d) + c use (a > 0)?

For monthly data with a period of 12 months, the trick θ = 30m works because

Why collect data for at least one full cycle?

Model h = 33 sin θ + 36. Predicted height at θ = 150°?

For parents and teachers

Big idea: Collecting data that can be modelled by a sine function, from primary and secondary sources, and graphing it (C3.1); deciding whether it is sinusoidal (C3.2).

How to help: Help with the hands-on part of the investigation (holding the bike steady, filming the pendulum, or finding a sunrise and sunset table). Ask your teen to explain how each number in their model comes from the data.

Safety: Keep fingers clear of bicycle spokes and chains; turn the wheel slowly by hand. Hang the pendulum where no one will walk into it.

Common difficulties:

Too few data points, so the shape is unclear.
Using the whole max-to-min distance as the amplitude.
Choosing d without checking where the data start (on the axis going up, at a maximum, or at a minimum).