Kenneth ChengMathematical modeling
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Applications of trigonometric functions

Read the idea, work independently, then explain what changed.

高一必修 第一册(A版).pdf · 5.7 · PDF 249 / printed page 242

Revisit first: The function y=A sin(ωx+φ)

TOPIC 01

Applications of trigonometric functions

Build understanding of applications of trigonometric functions through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use applications of trigonometric functions with explicit angle units and domains.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Geometry and units

Choose the appropriate side ratio and angular unit.

Periodic models

Interpret amplitude, midline, period and phase in the context.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in applications of trigonometric functions changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

θ+φ=45° = 0.7854 rad; sin=0.7071, cos=0.7071. The circle has radius 1.

θ+φ=45° = 0.7854 rad; sin=0.7071, cos=0.7071. The circle has radius 1.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

A 4-metre ladder makes 30° with level ground. Find the top height.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
Height is opposite the ground angle.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Calculate or simplify this relation.

    h=4sin⁡30∘=2h=4\sin30^{\circ}=2

  3. The ladder is the hypotenuse, not the horizontal distance.

The requested value is 2.

Checks and common pitfalls: The ladder is the hypotenuse, not the horizontal distance.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Find the period in hours.

h(t)=3+2sin⁡(πt/2)h(t)=3+2\sin(\pi t/2)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
One complete cycle changes the angle by 2π.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Calculate or simplify this relation.

    (π/2)T=2π⇒T=4(\pi/2)T=2\pi\Rightarrow T=4

  3. The period uses the same time unit as t.

The requested value is 4.

Checks and common pitfalls: The period uses the same time unit as t.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Two daily temperature observations fit a sinusoid. Explain why more observations are needed before predicting its period.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
Count the unknown parameters in a general sinusoid.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Record enough observations over several cycles and compare a fitted curve with residual errors and context.

  3. A visual fit alone does not establish a unique periodic model.

Several amplitudes, phases and periods can fit two observations.

Checks and common pitfalls: A visual fit alone does not establish a unique periodic model.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

A 6-metre ladder makes 30° with level ground. Find the top height.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
Height is opposite the ground angle.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Calculate or simplify this relation.

    h=6sin⁡30∘=3h=6\sin30^{\circ}=3

  3. The ladder is the hypotenuse, not the horizontal distance.

The requested value is 3.

Checks and common pitfalls: The ladder is the hypotenuse, not the horizontal distance.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Find the period in hours.

h(t)=3+2sin⁡(πt/3)h(t)=3+2\sin(\pi t/3)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
One complete cycle changes the angle by 2π.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Calculate or simplify this relation.

    (π/3)T=2π⇒T=6(\pi/3)T=2\pi\Rightarrow T=6

  3. The period uses the same time unit as t.

The requested value is 6.

Checks and common pitfalls: The period uses the same time unit as t.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

A wheel has radius 3 m and its centre is 5 m above ground. Find the greatest passenger height.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
At the top, add radius to centre height.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Calculate or simplify this relation.

    hmax⁡=(5)+3=8h_{\max}=(5)+3=8

  3. The corresponding minimum is 2 m, so the wheel remains above ground.

The requested value is 8.

Checks and common pitfalls: The corresponding minimum is 2 m, so the wheel remains above ground.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

At which first nonnegative time does the model reach its maximum?

h(t)=5+2sin⁡(πt/6)h(t)=5+2\sin(\pi t/6)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
Sine first reaches one at π/2.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Calculate or simplify this relation.

    πt/6=π/2⇒t=3\pi t/6=\pi/2\Rightarrow t=3

  3. Later maxima recur after a full period of 12.

The requested value is 3.

Checks and common pitfalls: Later maxima recur after a full period of 12.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

A tide has maximum 8 m and minimum 4 m. Find its mean level in a symmetric sinusoidal model.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
The midline bisects the extremes.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Calculate or simplify this relation.

    c=(8)+(4)2=6c=\frac{(8)+(4)}2=6

  3. This is a sinusoidal modelling assumption, not a claim about every real tide.

The requested value is 6.

Checks and common pitfalls: This is a sinusoidal modelling assumption, not a claim about every real tide.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

A student enters 30 into a calculator in radian mode to calculate sin 30°. Explain and repair the error.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
The numerical input must match the selected angular unit.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Calculate or simplify this relation.

    30∘=π/630^{\circ}=\pi/6

  3. Degrees and radians are measurements of the same angle with different scales.

Use degree mode or enter π/6 in radian mode; the value is 1/2.

Checks and common pitfalls: Degrees and radians are measurements of the same angle with different scales.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Could h(t)=1+2cos t model the height above ground of an object that must always stay above ground? Explain.

h(t)=1+2cos⁡th(t)=1+2\cos t
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Compare amplitude with the midline.
Hint 2
The amplitude exceeds the mean height.
Worked solution

  1. Compare amplitude with the midline.

  2. Calculate or simplify this relation.

    −1≤h(t)≤3;h(π)=−1-1\le h(t)\le3;\quad h(\pi)=-1

  3. A model may require a restricted time domain or a revised vertical shift.

Not over all real t, because the model reaches −1.

Checks and common pitfalls: A model may require a restricted time domain or a revised vertical shift.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

A 8-metre ladder makes 30° with level ground. Find the top height.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
Height is opposite the ground angle.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Calculate or simplify this relation.

    h=8sin⁡30∘=4h=8\sin30^{\circ}=4

  3. The ladder is the hypotenuse, not the horizontal distance.

The requested value is 4.

Checks and common pitfalls: The ladder is the hypotenuse, not the horizontal distance.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Find the period in hours.

h(t)=3+2sin⁡(πt/4)h(t)=3+2\sin(\pi t/4)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
One complete cycle changes the angle by 2π.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Calculate or simplify this relation.

    (π/4)T=2π⇒T=8(\pi/4)T=2\pi\Rightarrow T=8

  3. The period uses the same time unit as t.

The requested value is 8.

Checks and common pitfalls: The period uses the same time unit as t.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

A wheel has radius 4 m and its centre is 6 m above ground. Find the greatest passenger height.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Hint 2
At the top, add radius to centre height.
Worked solution

  1. Define the geometric or periodic quantities with units, then choose a trigonometric relation.

  2. Calculate or simplify this relation.

    hmax⁡=(6)+4=10h_{\max}=(6)+4=10

  3. The corresponding minimum is 2 m, so the wheel remains above ground.

The requested value is 10.

Checks and common pitfalls: The corresponding minimum is 2 m, so the wheel remains above ground.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for applications of trigonometric functions?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Use applications of trigonometric functions with explicit angle units and domains.
    • Geometry and units: Choose the appropriate side ratio and angular unit.
    • Periodic models: Interpret amplitude, midline, period and phase in the context.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Choose the appropriate side ratio and angular unit.
    • Expected reasoning: Interpret amplitude, midline, period and phase in the context.
    • Expected correction: Fitting a curve does not prove a physical law.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗