Lesson 4 of 5Trigonometric Functions

Unit 4 · Lesson 4 of 5

Modelling with Sinusoidal Functions

Build $y = a sin (k (x - d)) + c$ or $y = a cos (k (x - d)) + c$ models from descriptions and data — Ferris wheels, tides, daylight, temperature — then use them to predict values and to find when a given value occurs.

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Key things to remember

1Axis $c = \frac{max + min}{2}$ ; amplitude $∣ a ∣ = \frac{max - min}{2}$ .
2 $k = \frac{2 π}{period}$ , with the period in the model's time units.
3Start at a maximum: $+ cos$ ; at a minimum: $- cos$ ; on the axis and rising: $+ sin$ . Otherwise set $d$ to the time of a maximum and use $+ cos$ .
4To predict, substitute in radian mode; to find when a value occurs, isolate the trig function, take the inverse, find the second angle in the cycle, convert to time, then add periods.
5For cosine the second angle is $2 π - θ$ ; for sine it is $π - θ$ .
6Interpret every parameter with units: amplitude as radius or half-range, axis as average level, period as cycle length, d as the time of a key point.

Anything that repeats on a schedule is a candidate for a sinusoidal model. A seat on a Ferris wheel rises and falls once per revolution. The water in a harbour rises and falls roughly twice a day. Hours of daylight in Ontario swell to a peak near the June solstice and shrink to a low near the December solstice, every year. In each case you can measure a highest value, a lowest value, and how long one cycle takes, and those three measurements, plus a starting point, are everything a sinusoid needs. Once you have the equation, it answers two kinds of question: what the value will be at a given time, and when the value will reach a given level. In this course every model uses radians, so a full cycle corresponds to $2 π$ .

A large Ferris wheel with hanging gondolas, silhouetted against the evening sky, with a crowd of people walking in the foreground. — PhotoA Ferris wheel at the Canadian National Exhibition in Toronto. Each seat rises and falls once per turn, so a rider's height repeats in the pattern a sinusoidal function describes.Derrick Mealiffe, CC BY-SA 2.0, via Wikimedia Commons

From a description to an equation

Parameter	How to get it	Ferris wheel: diameter 40 m, axle 22 m up, 3 min per turn, board at the bottom
Axis c	Average of the maximum and minimum	$c = \frac{42 + 2}{2} = 22$ m (the axle height)
Amplitude \|a\|	Half the distance from minimum to maximum	$∣ a ∣ = \frac{42 - 2}{2} = 20$ m (the radius)
k	$k = \frac{2 π}{period}$	$k = \frac{2 π}{3}$ , with $t$ in minutes
Function and sign	Start at a max: $+ cos$ ; at a min: $- cos$ ; on the axis rising: $+ sin$	Starts at the bottom, so use $- cos$ and $d = 0$
Check	Substitute a known point	$t = 1.5$ : $- 20 cos π + 22 = 42$ m, the top, halfway round

Read every parameter from the situation, then test the finished equation on a point you did not use to build it.

h (t) = - 20 cos (\frac{2 π}{3} t) + 22

Fig 1The rider's height traced over time: the axle height is the axis, the radius is the amplitude, and one revolution is one period.

VideoSal builds a trig model for the height of a Ferris wheel rider from the wheel's size and speed, the same kind of problem as this section's worked example.Watch on YouTube

From data to an equation

Real data rarely arrives as a neat description, so you extract the same measurements from a table or graph. Suppose a harbour's water depth reaches a high of $8.6$ m at 3:00 a.m. and the following low of $1.4$ m at 9:12 a.m. The axis is $c = \frac{8.6 + 1.4}{2} = 5.0$ m and the amplitude is $a = \frac{8.6 - 1.4}{2} = 3.6$ m. High to low is half a cycle, $6.2$ h, so the period is $12.4$ h and $k = \frac{2 π}{12.4} \approx 0.507$ . Because the data begins at a high, cosine with a horizontal shift equal to the time of that high is the simplest choice: $d = 3$ . With data that is not perfectly sinusoidal, such as average monthly temperatures, use the largest and smallest values in the table, and expect the model to be close rather than exact. Graphing technology can also fit a sinusoidal regression to the data; compare its parameters with your hand-built ones as a check. Do not be surprised if a classmate's equation looks different from yours. The tide could equally be written with sine and a shift of $d = 3 - 3.1 = - 0.1$ (a quarter period before the high, when the water is at the axis and rising), or with $- cos$ and $d = 9.2$ , the time of the low. All three describe exactly the same curve, because shifting by a quarter or half period converts one form into another. What matters is that your model reproduces the data.

D (t) = 3.6 cos (\frac{2 π}{12.4} (t - 3)) + 5.0, t in hours after midnight

Fig 2Maximum, minimum, axis, amplitude and period: the five measurements you take from any data set before writing the model.

Predicting values and finding when a value occurs

Predicting is substitution: put the time into the model with your calculator in radian mode. Finding WHEN a value occurs runs the model backwards and is a trigonometric equation. Isolate the cosine (or sine), then apply the inverse function to get one angle. That single angle is not the whole answer. Cosine takes each value twice per cycle, so the second angle is $2 π$ minus the first; for sine it is $π$ minus the first. Then convert each angle back to time by dividing by $k$ (and adding $d$ ), and add whole periods to get later occurrences. Graphically, the same answers are the intersections of the model with the horizontal line $y =$ target value, which is a good way to see how many answers to expect before you calculate them.

- 20 cos (\frac{2 π}{3} t) + 22 = 30 \Rightarrow cos (\frac{2 π}{3} t) = - 0.4 \Rightarrow \frac{2 π}{3} t = 1.982 or 2 π - 1.982 = 4.301 \Rightarrow t \approx 0.946 or 2.054 min

Loading interactive…

Set the amplitude, period, shift and axis to match a context, then read off values and crossing points.

Interpreting the parameters in context

Parameter	Ferris wheel	Harbour tide	Daylight hours	Monthly temperature
Amplitude \|a\|	Wheel radius, 20 m	Half the tidal range, 3.6 m	How far day length swings from its average, about 3.25 h	Half the gap between the warmest and coldest months
Axis c	Axle height, 22 m	Mean water depth, 5.0 m	Average day length, about 12 h	The year's average temperature
Period	Time for one revolution, 3 min	About 12.4 h between high tides	One year, 365 days	12 months
Phase shift d	When the rider is first at the chosen start point	Time of the first high tide	Day of the spring equinox when using sine	Month of the coldest (or warmest) average

A parameter is only fully interpreted when it has units and a meaning in the situation, not just a number.

Radian mode, and k is not the period

With

k = \frac{2 π}{3}

, a calculator left in degree mode treats

cos (2.09)

as the cosine of about two degrees and every prediction is wrong without any warning. Set radian mode before you start. And remember

k

is how many radians pass per unit of time, not the period itself: a period of 12.4 h gives

k \approx 0.507

, not 12.4.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

A Ferris wheel has radius

15

m, its axle is

17

m above the ground, and it turns once every

2

minutes. A rider boards at the bottom at

t = 0

. Write

h (t)

, find the height after

0.75

min, and find when the rider first reaches

25

Example 2

The harbour depth is

D (t) = 3.6 cos (\frac{2 π}{12.4} (t - 3)) + 5.0

t

hours after midnight. A boat needs at least

4

m of water. During which part of the morning can it NOT use the harbour?

Example 3

In Toronto, day length is roughly

15.4

h at the June solstice and

8.9

h at the December solstice (approximate figures). Taking the spring equinox as day

80

, build a model, predict day

100

, and find when days are at least

14

h long.

Example 4

A southern Ontario city's average monthly temperature is about

- 5^{\circ}

C in January (month 1) and

2 2^{\circ}

C in July (month 7). Build a model and estimate when the monthly average is

1 5^{\circ}

Where marks get lost

Using the period as $k$ : a 12.4 h tide needs $k = \frac{2 π}{12.4} \approx 0.507$ , not $12.4$ .
Evaluating the model with the calculator in degree mode, which gives wrong values with no warning.
Reporting only the calculator's first inverse-cosine answer and missing the second time in the cycle.
Forgetting to undo the horizontal shift: after dividing by $k$ , you must still add $d$ to get the actual time.
Using +cos for a Ferris wheel when the rider boards at the bottom, which puts the rider at the top at t = 0.

Flashcards 9

Axis of a sinusoidal model

Amplitude of a sinusoidal model

k

from the period

Model starts at a minimum (rider boards at the bottom)

+ 5 more

Practice

A tide has a period of

12.4

h. Find

k

for a model in radians, to 3 decimal places.

For

h (t) = - 20 cos (\frac{2 π}{3} t) + 22

, find the height in metres at

t = 1

min.

A Ferris wheel has diameter 30 m, its axle is 17 m above the ground, and it turns once every 2 min. A rider boards at the bottom. Which model gives the height?

Monthly average temperatures range from

- 5^{\circ}

C to

2 2^{\circ}

C. What is the axis

c

of the model, in degrees Celsius?

°C

High tide is

8.6

m and low tide is

1.4

m. Find the amplitude, in metres.

For

h (t) = - 15 cos (π t) + 17

, when does the rider first reach

25

m? Answer in minutes to 2 decimal places.

min

During each full revolution, how many times is a Ferris wheel rider at a height strictly between the lowest and highest points, such as 30 m on a wheel from 2 m to 42 m?

In a Ferris wheel model, which parameter letter represents the height of the axle?

Put the steps for finding when a sinusoidal model reaches a value in order.

1Divide by k and add d to convert angles to times
2Find the second angle in the same cycle
3Add whole periods for later occurrences
4Set the model equal to the target value
5Isolate the sine or cosine
6Take the inverse to get the first angle

Match each part of

D (t) = 3.6 cos (\frac{2 π}{12.4} (t - 3)) + 5.0

to its meaning.

3.6

12.4

5.0

Using

D (t) = 3.25 sin (\frac{2 π}{365} (t - 80)) + 12.15,

predict the day length on day

100

, in hours to 1 decimal place.

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