B. Trigonometric Functions · Radian Measure and Trigonometric Functions

Modelling Periodic Phenomena with Sinusoidal Functions

B2.7 B2.6About 75 minutes

Learning goals

I can build a sinusoidal model, with time in the role of the radian input, for a Ferris wheel, tides, hours of daylight or a spring.
I can explain what each parameter means in the situation, with units.
I can use a model to predict values, giving exact answers when the input gives a special angle.
I can pose my own questions about a periodic situation and state a sensible domain and range.

Words to know

mathematical model
An equation (or graph) that describes a real situation closely enough to make predictions.

independent variable
The input of the model, usually time t (in seconds, hours, days or months).

angular speed
How fast something turns, in radians per unit of time. For a period T it is 2π/T, which is the value of k.

domain of a model
The input values that make sense in the situation, for example 0 ≤ t ≤ 600 for a 10-minute ride.

semidiurnal tide
A tide pattern with two high tides and two low tides each day, a period of about 12.4 hours.

Why radians make modelling easy

In real situations the input is almost never an angle. It is time. With radians, the input of sin and cos is just a number, so a model like h(t) = −20 cos(πt/40) + 22 takes t in seconds directly. The value k = 2π ÷ (period) converts time into "how far around the cycle" we are.

Remember from Grade 11: in Sinusoidal Functions you modelled Ferris wheels, tides and daylight with degrees. Now you will write the same kinds of models in radians, the way scientists and engineers do.

A modelling routine:

Name the variables and units (t in seconds, h in metres...).
Find the max and min → c and |a|.
Find the period → k = 2π ÷ period.
Decide where a cycle starts at t = 0 (or another known time) → sine or cosine, the sign of a, and d.
Write the model, check it with a known point, and state a sensible domain.

A Ferris wheel with radius 20 m and centre 22 m above the ground (not to scale). A rider boards at the bottom, so the height starts at its minimum.

Worked example

Example 1: The Ferris wheel

A Ferris wheel has a radius of 20 m, its centre is 22 m above the ground, and it turns once every 80 s. A rider boards at the lowest point at t = 0. Write a model for the rider's height h (m) after t seconds, then find the height at t = 10 s and t = 30 s exactly.

Max 22 + 20 = 42 m, min 22 − 20 = 2 m, so c = 22 and |a| = 20.
Period 80 s, so k = 2π/80 = π/40.
At t = 0 the rider is at the minimum: use a reflected cosine with d = 0. h(t) = −20 cos(πt/40) + 22.
t = 10: −20 cos(π/4) + 22 = −20(√2/2) + 22 = 22 − 10√2 ≈ 7.9 m.
t = 30: −20 cos(3π/4) + 22 = −20(−√2/2) + 22 = 22 + 10√2 ≈ 36.1 m.

h(t) = −20 cos(πt/40) + 22; h(10) = 22 − 10√2 ≈ 7.9 m; h(30) = 22 + 10√2 ≈ 36.1 m.

The model h(t) = −20 cos(πt/40) + 22 for two rotations. Minimum 2 m at t = 0 and 80 s; maximum 42 m at t = 40 and 120 s.

Worked example

Example 2: Tides

On the Bay of Fundy coast, which has some of the highest tides in the world, the depth of water at a dock is modelled with a period of 12.4 h. Suppose high tide is 7.2 m at 2:00 a.m. (t = 2) and low tide is 1.0 m. Write a model and predict the depth at 5:00 a.m.

c = (7.2 + 1.0)/2 = 4.1; a = (7.2 − 1.0)/2 = 3.1.
k = 2π/12.4 ≈ 0.507 (keep it as 2π/12.4).
High tide at t = 2: cosine with d = 2. h(t) = 3.1 cos((2π/12.4)(t − 2)) + 4.1.
At t = 5: 3.1 cos(2π(3)/12.4) + 4.1 = 3.1 cos(1.520) + 4.1 ≈ 3.1(0.0507) + 4.1 ≈ 4.26.

h(t) = 3.1 cos((2π/12.4)(t − 2)) + 4.1; at 5:00 a.m. the depth is about 4.3 m. (Low tide comes half a period after high tide, at t = 8.2, about 8:12 a.m.)

Worked example

Example 3: Hours of daylight in Toronto

In Toronto, the longest day (near June 20) has about 15.4 h of daylight and the shortest (near December 21) about 9.0 h. Daylight is 12.2 h halfway between, near March 21 (day 80 of the year). Write a model for the hours of daylight D on day t, and predict D on July 1 (day 182).

c = (15.4 + 9.0)/2 = 12.2; a = (15.4 − 9.0)/2 = 3.2.
Period 365 days: k = 2π/365.
On day 80 the daylight is at the axis and increasing: a sine with d = 80.
D(t) = 3.2 sin((2π/365)(t − 80)) + 12.2.
D(182) = 3.2 sin(2π(102)/365) + 12.2 ≈ 3.2 sin(1.756) + 12.2 ≈ 3.2(0.983) + 12.2 ≈ 15.3.

D(t) = 3.2 sin((2π/365)(t − 80)) + 12.2; on July 1, about 15.3 h of daylight. The model's maximum comes at t = 80 + 365/4 ≈ 171, about June 20 ✓.

The daylight model for Toronto over one year. It crosses its axis (12.2 h) going up near March 21, peaks near June 20 and is lowest in late December.

Worked example

Example 4: A spring

A mass on a spring is pulled 6 cm below its rest position and released at t = 0. It bounces with a period of 0.8 s. Write a model for its displacement d (cm, positive above rest) and find d at t = 0.3 s.

Rest position is the axis: c = 0. Amplitude 6 cm.
k = 2π/0.8 = 5π/2.
It starts at the lowest point: d(t) = −6 cos(5πt/2).
d(0.3) = −6 cos(0.75π) = −6 cos(3π/4) = −6(−√2/2) = 3√2 ≈ 4.2.

d(t) = −6 cos(5πt/2); at 0.3 s the mass is 3√2 ≈ 4.2 cm above rest. (A real spring slowly loses amplitude; the model ignores that, which is called damping. Damped models appear in Combining Functions and Modelling.)

Posing problems

Pose your own problems. Good modelling questions ask: "What is the value at time …?", "When does it first reach …?", "How long is it above …?" and "How fast is it changing at …?". The next lesson answers the last three kinds.

Just for fun

A Ferris wheel is really just a giant unit circle with seats. The math is the same; the snacks are better.

Hands-on activity

Pendulum model

You need: a set of keys (or a small bag of rice) tied securely to a string about 50 cm long, a ruler, a timer (phone stopwatch), and a doorway or a hook to hang it from. Ask a grown-up to help you hang it safely, at a height where nobody will walk into it.

Let the keys hang still. That is the rest position.
Pull them 10 cm to one side (measure along the floor direction with the ruler) and let go at t = 0.
Time 10 full swings (out and back). Divide by 10 to get the period T.
Write a model for the sideways displacement: d(t) = 10 cos(2πt/T), in centimetres.
Use your model to predict where the keys are at t = T/4, T/2 and 3T/4. Watch closely to check.
Pose a question of your own, such as "Where are the keys after 5 seconds?", and answer it with the model.

What to look for: for a 50 cm string the period is about 1.4 s, so k = 2π/1.4 ≈ 4.5. At T/4 and 3T/4 the keys pass through the rest position (d = 0), and at T/2 they are 10 cm out on the other side (d = −10). The swings slowly get smaller; your model ignores that.

Your turn

A Ferris wheel has a diameter of 30 m, its centre is 17 m above the ground, and it turns once every 60 s. A rider boards at the bottom at t = 0. Write a model for the height h (m).

Use your model to find the height at t = 15 s.

Find the height at t = 20 s exactly.

The model h(t) = 2.5 cos((2π/12.4)(t − 3)) + 4 gives the water depth in metres at a dock, t hours after midnight. State the period and when and how high the first high tide after midnight is.

For the same tide model, when is the next low tide?

A mass on a spring moves 4 cm above and below its rest position with a period of 2 s. It is released from 4 cm above rest at t = 0. Write a model.

A Ferris wheel ride lasts exactly three rotations of 60 s each. State a sensible domain for the model in the first question.

More practice: warm-up

A height varies between 10 m and 50 m. What is the axis?

The period is 80 s. Find k.

A rider starts at the bottom. Sine, cosine or reflected cosine?

A tide has period 12.4 h. Find k to three decimals.

What is the period of y = 3 sin(πt/6) + 10?

More practice: on your own

For the daylight model D(t) = 3.2 sin((2π/365)(t − 80)) + 12.2, state the maximum and minimum hours of daylight and the day of the maximum.

Use the daylight model to predict the hours of daylight on day 355 (December 21).

Pose and answer a question about the Ferris wheel in Example 1 that uses t = 60 s.

For the spring d(t) = −6 cos(5πt/2), find d at t = 0.2 s and explain what is happening.

In the Ferris wheel model h(t) = −20 cos(πt/40) + 22, which parameter changes, and to what, if (a) the wheel turns once every 60 s, (b) the whole wheel is built on a platform 3 m higher?

Challenge

Write a sine model for the Ferris wheel in Example 1, and check that it gives 2 m at t = 0.

A wheel has radius 25 m and centre 27 m above the ground, and turns once every 2 min. At t = 0 a rider is level with the centre and moving up. Write a model (t in seconds) and find the exact height at t = 100 s.

A boat needs at least 3 m of water at the dock in the tide model h(t) = 2.5 cos((2π/12.4)(t − 3)) + 4. Can it dock at t = 8? Explain.

Check your understanding

Quiz

A Ferris wheel has radius 12 m, its centre is 14 m above the ground and it turns once every 90 s. A rider boards at the bottom at t = 0. Which model fits?

A tide has a period of 12.4 h. To three decimals, k is about...

In D(t) = 3.2 sin((2π/365)(t − 80)) + 12.2, what does 12.2 represent?

What is the period of d(t) = 5 cos(4πt), t in seconds?

What is the greatest height given by h(t) = −20 cos(πt/40) + 22?

A model describes one 10-minute ride, with t in seconds. Which domain makes sense?

For parents and teachers

Big idea (B2.7): periodic situations (Ferris wheels, tides, daylight, springs) can be modelled with y = a sin(k(t − d)) + c or the cosine version, where t is time and k = 2π ÷ period. Each parameter has a real meaning: c is the average level, |a| the swing above and below it, and d sets the timing.

The Toronto daylight numbers (about 15.4 h and 9.0 h) and the tide numbers are rounded, model-sized values for practice, not official data.

How to help: Ask your teen to explain each number in a model in plain words ("What does the 22 mean for the Ferris wheel?"). Do the pendulum activity together and help with the timing.

Common difficulties:

Using k = period instead of 2π ÷ period.
Choosing +cos when the situation starts at a minimum.
Leaving the calculator in degree mode (the models only work in radians).
Forgetting units and a sensible domain.