D. Trigonometric Functions · Sinusoidal Functions

Modelling Periodic Phenomena That Do Not Involve Angles

D3.3 D3.2About 75 minutes

Learning goals

I can explain how a sinusoidal function can model something that changes with time rather than with an angle.
I can write a sinusoidal model for a Ferris wheel, tides, daylight or a population from a description.
I can use a model to calculate values and interpret each parameter in context.
I can state reasonable domain and range restrictions for a model.

Words to know

independent variable
The input of a model, such as time t (seconds, hours, days).

k as a conversion factor
In a time model, k = 360 ÷ period changes time units into degrees, so that one period of time matches one 360° cycle.

tide
The regular rise and fall of the ocean, caused mostly by the Moon's gravity; at many Canadian harbours two high tides happen about every 24 h 50 min.

in context
Interpreted in terms of the real situation, with units.

What if there is no angle?

The tide at a harbour does not turn through an angle. Neither do the hours of daylight or the number of snowshoe hares in a forest. Yet all of them rise and fall in smooth cycles. Can sine and cosine still describe them? Yes. The trick is to let the input be time, and to use k to turn time into degrees.

Time in, degrees inside

In a model such as h = a cos(k(t − d)) + c, the variable t is a time (seconds, hours, days, years). The calculator still needs degrees inside the cosine. That is the job of k:

k = 360 ÷ period

When t increases by one full period, k(t − d) increases by exactly 360°, so the cosine completes exactly one cycle. For example, tides with a period of 12.4 h use k = 360 ÷ 12.4 ≈ 29.0: the inside of the function grows by about 29° for every hour.

For a Ferris wheel, k(t − d) really is the angle the wheel has turned. For tides or daylight it is not a real angle at all; it is just a clock that runs from 0° to 360° once per cycle. That is why sinusoidal functions can model so many things that do not involve angles.

A Ferris wheel with radius 20 m and axle 22 m above the ground, turning once every 60 s, with the rider boarding at the bottom. Each label gives the time t and the rider's height. Here 6t is the real angle turned (not to scale).

Worked example

Example 1: A Ferris wheel model

Write a model for the rider's height h (m) after t seconds for the Ferris wheel in the diagram, and find the height after 10 s and after 50 s.

c = axle height = 22; |a| = radius = 20; period 60 s, so k = 360 ÷ 60 = 6.
The rider starts at the bottom (a minimum) at t = 0, so use a reflected cosine with d = 0: h(t) = −20 cos(6t) + 22.
h(10) = −20 cos 60° + 22 = −20(0.5) + 22 = 12 m.
h(50) = −20 cos 300° + 22 = −20(0.5) + 22 = 12 m. (10 s after boarding and 10 s before getting back to the bottom: the same height, by symmetry.)
Another correct model: the rider passes the axle height going up at t = 15 s, so h(t) = 20 sin(6(t − 15)) + 22.

h(t) = −20 cos(6t) + 22; 12 m at both times.

Tides

At many Canadian harbours there are two high tides and two low tides a day, and the time from one high tide to the next is about 12.4 hours (12 h 25 min). The Bay of Fundy, between New Brunswick and Nova Scotia, is famous for having some of the largest tides on Earth.

Worked example

Example 2: A tide model

Sample data for one day at a harbour on the Bay of Fundy: high tide is 13 m at 4:30 a.m. and low tide is 1 m. The period is 12.4 h. Write a model for the water height h (m), t hours after midnight, and find the height at noon.

c = (13 + 1) ÷ 2 = 7; a = (13 − 1) ÷ 2 = 6; k = 360 ÷ 12.4 ≈ 29.0.
A maximum at t = 4.5, so use cosine with d = 4.5: h = 6 cos(29.0(t − 4.5)) + 7.
At noon, t = 12: 29.0(12 − 4.5) ≈ 217.7°. cos 217.7° ≈ −0.791. h ≈ 6(−0.791) + 7 ≈ 2.3 m.
Interpretation: at noon the tide is low, about 1.3 m above its lowest level, and rising (low tide was at 4.5 + 6.2 = 10.7, about 10:42 a.m.).

h = 6 cos(29.0(t − 4.5)) + 7; about 2.3 m at noon.

The tide model over one day. High tides are 12.4 h apart (4:30 a.m. and 4:54 p.m.), so the tides come about 25 minutes later each half day.

Worked example

Example 3: Interpreting a daylight model

The hours of daylight D in a southern Ontario city on day n of the year are modelled by D(n) = 3.2 sin(0.986(n − 80)) + 12.2. (a) Explain each number. (b) Find the daylight on Canada Day (July 1, day 182).

0.986 ≈ 360 ÷ 365: the period is one year of 365 days.
12.2 is the axis: on average there are about 12.2 h of daylight. 3.2 is the amplitude: daylight goes up to 12.2 + 3.2 = 15.4 h and down to 12.2 − 3.2 = 9.0 h.
80 is the phase shift: day 80 (March 21), near the spring equinox, is where the sine starts its cycle: daylight is at its average and increasing. The maximum comes a quarter of a year later, at day 80 + 91.25 ≈ 171 (June 20).
(b) D(182) = 3.2 sin(0.986 × 102) + 12.2 = 3.2 sin(100.6°) + 12.2 ≈ 3.2(0.983) + 12.2 ≈ 15.3 h.

About 15.3 hours of daylight on Canada Day.

Domain and range in context

A model's equation works for every real t, but the situation does not. A 10-minute Ferris wheel ride has domain 0 ≤ t ≤ 600 (seconds). A daylight model for one year uses 1 ≤ n ≤ 365. A population model cannot give negative numbers of animals, and a tide model only gives typical heights (storms and winds change the real tide). Always ask: for which inputs does this model make sense, and what outputs are possible?

Just for fun

In parts of the Bay of Fundy, the difference between high and low tide can be taller than a three-storey building. At some harbours, fishing boats sit on the mud at low tide and float again a few hours later, so knowing your sinusoidal model really can decide when you get to go home.

Hands-on activity

Predict your paper-plate Ferris wheel

You need: the paper-plate Ferris wheel from the first lesson of this unit (or make a new one: a paper plate pinned through its centre to cardboard taped upright), a ruler and a calculator.

Measure the plate's radius r and the height H of the pin above the counter.
Let n be the number of eighth-turns from the bottom. One full turn is 8 eighth-turns, so k = 360 ÷ 8 = 45. Write the model h = −r cos(45n) + H.
Predict the height of the dot after n = 3 and after n = 5 eighth-turns.
Turn the plate and measure those heights. How close were your predictions?
Rewrite the model for a rider who boards at the top instead.

What to look for: with r = 11 cm and H = 15 cm, h = −11 cos(45n) + 15 predicts 15 + 11(0.707) ≈ 22.8 cm for both n = 3 and n = 5 (they are mirror positions). Measurements within a few millimetres mean the model works. A rider who boards at the top has h = r cos(45n) + H.

Your turn

A Ferris wheel has radius 10 m, its axle is 12 m above the ground, and it turns once every 40 s. A rider boards at the bottom at t = 0. Write a model for the rider's height.

Use your model from the previous question to find the height at t = 10 s and at t = 25 s.

In h = −10 cos(9t) + 12, what does 9t represent? Would it have the same meaning in a tide model?

For the tide model h = 6 cos(29.0(t − 4.5)) + 7, when is the next high tide after 4:30 a.m.?

In D(n) = 3.2 sin(0.986(n − 80)) + 12.2, what do 12.2 and 3.2 mean in context?

Use the daylight model to find the daylight on day 355 (December 21).

Snowshoe hare populations in parts of Canada are known to rise and fall in cycles of about 10 years. A sample model is P = 20 000 sin(36(t − 2)) + 30 000, with t in years. State the period, the maximum and the minimum population.

More practice: warm-up

A daily cycle has a period of 24 h. Find k.

A yearly cycle has a period of 365 days. Find k to three decimal places.

In h = −20 cos(6t) + 22, how high is the axle of the Ferris wheel?

If t is in hours, what are the units of k?

A Ferris wheel rider boards at the bottom. Which is the simplest choice of function: sine, cosine or reflected cosine?

More practice: on your own

A Ferris wheel 50 m in diameter has its lowest point 2 m above the ground and turns once every 3 min. A rider boards at the bottom. Write a model for the height h (m) after t seconds and find h(60).

A town's average monthly temperature is highest in July (m = 7) at 20 °C and lowest in January at −10 °C. Write a model and predict the average October temperature.

High tide at a harbour is 8 m at 1:00 a.m. and low tide is 2 m at 7:12 a.m. Write a model and find the height at 4:06 a.m.

The Ferris wheel h(t) = −20 cos(6t) + 22 gives a ride of 4 complete turns. State the domain and range of the model in context.

A buoy bobs on ocean waves. Its highest and lowest points are 0.6 m apart, a wave passes every 4 s, and it is at its highest point at t = 0. Write a model for its height h (m) relative to the calm water level.

Challenge

The Ferris wheel from Example 1 (radius 20 m, axle 22 m, 60 s per turn) loads riders from a platform level with the axle, on the side where the wheel is going up. Write a model using sine and one using cosine.

Sample data for a northern community: about 20 h of daylight on day 172 and about 4 h on day 355. Write a cosine model and use it to estimate the daylight on day 80.

A tide model uses t in hours with period 12.4 h. Rewrite k so that t is measured in minutes instead. Then explain how the model changes.

Check your understanding

Quiz

A tide has a period of 12.4 h. In a model with t in hours, what is k?

A Ferris wheel has radius 15 m, axle 18 m above the ground and period 30 s. A rider boards at the bottom. Which model fits?

In D(n) = 3.2 sin(0.986(n − 80)) + 12.2, what does 12.2 represent?

What is the period of P = 20 000 sin(36(t − 2)) + 30 000 (t in years)?

What is a sensible domain for a model of a 10-minute Ferris wheel ride, with t in seconds?

In a time-based model h = a cos(k(t − d)) + c, what is the job of k?

For parents and teachers

Big idea: D3.3 asks students to see how sinusoidal functions can model periodic phenomena that do not involve angles: time, days, years. The key idea is that k = 360 ÷ period converts time into degrees. D3.2 (domain and range restrictions in context) is practised throughout.

How to help: Pick a real situation together (tides at a place you have visited, the daylight where you live, a Ferris wheel at a local fair) and ask your teen to explain what each of a, k, d and c would mean.

Common difficulties:

Using the period itself for k (k = 12.4 instead of 360 ÷ 12.4).
Converting clock times: 4:30 a.m. is t = 4.5, not 4.3; 0.4 h is 24 min, not 40 min.
Choosing the wrong starting function. A rider at the bottom needs −cos; at the axis going up, sin.
Rounding k too early. Keep k to at least three significant figures (29.0, 0.986), or use the exact fraction on the calculator.