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C. Trigonometric Functions · Sine Functions

Modelling Projects: Posing and Solving Sine Problems

C3.3 C3.1 C2.2About 75 minutes
Learning goals
  • I can follow the modelling cycle from a question to a checked answer.
  • I can pose different types of questions that a sine model can answer.
  • I can build sine models from published data and use them to compare situations.
  • I can solve design problems by choosing a, c and d to meet conditions.

Words to know

modelling cycle
The process of asking a question, collecting data, graphing, building a model, solving, and checking and refining.
pose a problem
To write a question that can be answered using a model or data.
alternating current (AC)
Electric current that reverses direction regularly; in Canadian homes it cycles 60 times per second.
hertz (Hz)
A unit meaning cycles per second.

Your turn to ask the questions

In Lesson 13 the questions were given to you. Real modelling starts earlier: someone notices a periodic pattern, asks a question about it, finds data, builds a model, and uses it. In this lesson you will practise the whole cycle, including posing good questions, with three Canadian contexts: daylight in the North and the South, Ferris wheel design, and the electricity in your home.

1. Ask a question2. Collect or find data3. Graph it4. Build a model5. Solve and answer6. Check and refinemodelling cycle
The modelling cycle. After checking, you often go around again with a better question or a better model.

What makes a good question?

A good question can be answered with the model and matters in the context. Most fall into a few types:

  • Find an output: "How much daylight on May 21?"
  • Find an input: "When does the seat first reach 30 m?"
  • How long: "For how many months is there more than 16 h of daylight?"
  • Compare: "When does Iqaluit have more daylight than Windsor?"
  • Predict (extrapolate): "What will the depth be at noon tomorrow?"
  • Design: "What radius and centre height give a ride from 2 m to 50 m?"

Context 1: Daylight in Iqaluit and Windsor

Iqaluit, the capital of Nunavut, is far north (about 64° N). Windsor, Ontario, is one of the most southerly cities in Canada (about 42° N). Approximate hours of daylight on the 21st of each month (calculated from the sun's position and rounded):

MonthJanFebMarAprMayJunJulAugSepOctNovDec
Iqaluit (h)6.09.212.215.618.820.819.115.812.59.46.14.3
Windsor (h)9.610.812.113.614.815.314.813.712.310.99.69.1
Approximate hours of daylight on the 21st of each month.
Worked example

Example 1: Building two models

Using θ = 30m degrees (m = month number, January = 1), build a model D = a sin(30m − d) + c for each city.

  1. Iqaluit: max 20.8 (June), min 4.3 (December). c = (20.8 + 4.3) ÷ 2 = 12.55, a = (20.8 − 4.3) ÷ 2 = 8.25. The max is in June (m = 6, θ = 180°), so d = 180° − 90° = 90°.
  2. Model: D = 8.25 sin(30m − 90°) + 12.55.
  3. Windsor: max 15.3, min 9.1. c = 12.2, a = 3.1, and the max is also in June, so d = 90°.
  4. Model: D = 3.1 sin(30m − 90°) + 12.2.
  5. Check Iqaluit in April (m = 4): 8.25 sin 30° + 12.55 ≈ 16.7 h; table 15.6 h. The model is about 1 h too high in spring, because real daylight in the far North does not follow a perfect sine curve. It is still useful for estimates.

Iqaluit: D = 8.25 sin(30m − 90°) + 12.55. Windsor: D = 3.1 sin(30m − 90°) + 12.2.

mhours012345678910111204812162024Iqaluit modelWindsor model
Data (dots) and models for Iqaluit (solid, blue dots) and Windsor (dashed, yellow dots). Both have a period of 12 months and a maximum in June, but Iqaluit's amplitude is much larger.
Worked example

Example 2: Posing and answering questions

Pose three questions about the two daylight models, of different types, and answer them.

  1. Q1 (compare amplitudes): "How much more does daylight swing in Iqaluit than in Windsor?" Amplitudes 8.25 h and 3.1 h; Iqaluit's is about 2.7 times as large (8.25 ÷ 3.1 ≈ 2.66).
  2. Q2 (how long): "For how many months does Iqaluit have more than 16.675 h of daylight, according to the model?" Solve 8.25 sin(30m − 90°) + 12.55 = 16.675 → sin(30m − 90°) = 0.5 → 30m − 90° = 30° or 150° → m = 4 or m = 8. Above between them: about 4 months (April 21 to August 21 in the model).
  3. Q3 (compare): "When does Iqaluit have more daylight than Windsor?" Both models are above their axes from the March equinox to the September equinox, and Iqaluit's axis (12.55) is very close to Windsor's (12.2). From the table, Iqaluit has more from about the March equinox (around March 20) to late September (March: 12.2 vs 12.1, September: 12.5 vs 12.3), and less in the other months.

Sample answers: amplitude about 2.7 times as large; about 4 months above 16.675 h; Iqaluit has more daylight from about the March equinox to late September.

Context 2: Designing a Ferris wheel

Design questions work backwards: you choose the parameters so that the model meets conditions. For a wheel boarded at the bottom, h(θ) = a sin(θ − 90°) + c with a = radius and c = centre height.

Worked example

Example 3: A design problem

A fair wants a Ferris wheel whose seats board 1.5 m above the ground and reach a top height of 41.5 m. (a) Find the model. (b) A safety rule says riders should not be above 35 m for more than one-third of each turn. Does this wheel meet the rule?

  1. (a) Radius a = (41.5 − 1.5) ÷ 2 = 20 m; centre c = (41.5 + 1.5) ÷ 2 = 21.5 m. Model: h = 20 sin(θ − 90°) + 21.5.
  2. (b) Solve 20 sin(θ − 90°) + 21.5 = 35 → sin(θ − 90°) = 0.675 → θ − 90° ≈ 42.5° or 137.5° → θ ≈ 132.5° or 227.5°.
  3. Above 35 m from about 132.5° to 227.5°, which is 95° of each 360° turn. 95 ÷ 360 ≈ 0.26, which is less than one-third (0.33). The wheel meets the rule.

h = 20 sin(θ − 90°) + 21.5; riders are above 35 m for about 26% of each turn, so the rule is met.

Context 3: The electricity in your home

The electrical outlets in Canadian homes supply alternating current (AC). The voltage is sinusoidal: it swings positive and negative 60 times per second (60 Hz). Outlets are rated at 120 V, which is a kind of average; the voltage actually swings between about +170 V and −170 V.

With t in seconds, one cycle takes 1/60 s. In degrees, the angle must go 360° × 60 = 21 600° each second, so a model is

V(t) = 170 sin(21 600t)

Safety: never put anything into an outlet or open electrical equipment. The voltage and current from an outlet can kill. This model is for calculation only.

t (s)V (volts)01/1201/601/401/30−200−1000100200peak ≈ 170 V≈ −170 V
Two cycles of household AC voltage, V(t) = 170 sin(21 600t). The whole picture lasts only 1/30 of a second.
Worked example

Example 4: Questions about AC voltage

Use V(t) = 170 sin(21 600t). (a) What is the period? (b) What is the voltage at t = 1/240 s? (c) How many times per second is the voltage exactly 0?

  1. (a) Period = 360 ÷ 21 600 = 1/60 s (about 0.0167 s).
  2. (b) 21 600 × (1/240) = 90°, so V = 170 sin 90° = 170 V (a peak).
  3. (c) A sine wave crosses zero twice per cycle. 60 cycles per second × 2 = 120 times per second.

(a) 1/60 s (b) 170 V (c) 120 times per second.

Just for fun

An electrician walks into a math class and says, "I work with sine waves all day."

The teacher replies, "Current events are my favourite topic."

Hands-on activity

Pose and solve your own sine problem

You need: one of the models from this unit (or the model you built in Lesson 12), paper, a pencil, a calculator and (optional) a graphing app. A family member can play the "client".

  1. Choose a context: daylight in your town, a Ferris wheel, a tide, or your Lesson 12 data.
  2. Write three questions of different types from the list above (for example: find an output, how long, design).
  3. Have your family member pick one question and ask you to answer it like a consultant: explain the model, solve the question, and give the answer in a sentence with units.
  4. Check the answer by substitution or with a graph, and name one way the real situation might differ from the model.

What to look for: each question can actually be answered with the model; answers have units and are in the domain; the explanation mentions the meaning of a, c and d; the limitation is reasonable (weather, rounding, the curve not being a perfect sine wave, a ride stopping to let people on).

Your turn

1

Using D = 3.1 sin(30m − 90°) + 12.2, find Windsor's predicted daylight on May 21 (m = 5).

2

Using D = 8.25 sin(30m − 90°) + 12.55, find Iqaluit's predicted daylight on December 21.

3

Write one 'find an input' question about the Ferris wheel h = 20 sin(θ − 90°) + 21.5, and answer it.

4

For V(t) = 170 sin(21 600t), find V at t = 1/120 s.

5

A wheel must board at 2 m and reach 62 m. Find a, c and the model (boarding at the bottom).

6

Classify this question by type: "How many hours of daylight will Windsor have on June 21 next year?"

More practice: warm-up

1

What is the period of the daylight models?

2

Which city's daylight model has the greater amplitude?

3

How many cycles per second does Canadian household AC have?

4

For a wheel boarded at the bottom, what does c mean?

5

Name one step of the modelling cycle that comes after building a model.

More practice: on your own

1

Using Windsor's model, for how many months is daylight above 13.75 h? (Hint: solve D = 13.75.)

2

Using Iqaluit's model, find the months in which daylight is less than 8.425 h.

3

A smaller wheel: h = 6 sin(θ − 90°) + 7.5. For what fraction of each turn is the seat above 10.5 m?

4

Explain why the Iqaluit model is less accurate in spring than the Windsor model.

5

Pose a 'design' question for a tide model d(t) = a sin(30t) + c and answer it.

Challenge

1

Use the two daylight models to find when Iqaluit and Windsor have equal daylight. (Set the models equal and solve for m.)

2

The fair's safety rule changes: riders may be above 35 m for at most one-quarter of each turn. Find the largest top height a wheel boarding at 1.5 m can have. (Hint: at the limit, the seat is above 35 m for exactly 90°, from θ = 135° to θ = 225°.)

3

For V(t) = 170 sin(21 600t), how long in each cycle is the voltage above 85 V?

Check your understanding

Quiz

1.

Which question is a 'how long' question?

2.

Iqaluit's daylight model has amplitude 8.25 h and axis 12.55 h. What is the predicted maximum?

3.

A wheel boards at 1 m and its top is at 31 m. Which model fits (boarding at the bottom)?

4.

Canadian household AC voltage completes how many cycles per second?

5.

For V(t) = 170 sin(21 600t), what is the period?

6.

Which is the best final step after solving a modelling problem?

For parents and teachers

Big idea: Posing and solving problems with sine models (C3.3), using collected or published data (C3.1) and predicting periodic behaviour (C2.2).

How to help: Be the "client": ask your teen a real question ("When should we go to the beach to see low tide?" "How much daylight will we have on your birthday?") and have them answer it with a model and explain it to you.

Safety: The household electricity example is for calculation only. Never let anyone experiment with outlets or open electrical devices.

Common difficulties:

  • Writing questions that the model cannot answer (for example, about the weather on a specific day).
  • Forgetting that a model is approximate; encourage your teen to compare with real data and say how close it is.
  • Mixing up the type of answer needed: an angle, a time, a height or a length of time.