Kenneth ChengMathematical modeling
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Graphs and properties of trigonometric functions

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

高一必修 第一册(A版).pdf · 5.4 · PDF 203 / printed page 196

Revisit first: Reduction formulas

TOPIC 01

Graphs and properties of trigonometric functions

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

What you will be able to explain

  • Use graphs and properties 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.

Cycle and range

Sine and cosine repeat every 2π and take values from −1 to 1.

Tangent restrictions

Tangent excludes zeros of cosine and has period π.

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 graphs and properties 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

The least positive period is cπ. Find c.

y=sin⁡(2x)y=\sin(2x)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Hint 2
The input angle must increase by one complete cycle.
Worked solution

  1. Read amplitude, period and allowed intervals directly from the trigonometric graph.

  2. Calculate or simplify this relation.

    2T=2π⇒T=22π2T=2\pi\Rightarrow T=\frac2{2}\pi

  3. Frequency multiplies the input and divides the period.

The requested value is 1.

Checks and common pitfalls: Frequency multiplies the input and divides the period.

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

02 / Standard#Worked example

State the full range.

y=2cos⁡x+1y=2\cos x+1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Hint 2
Scale the cosine bounds and shift upward.
Worked solution

  1. Read amplitude, period and allowed intervals directly from the trigonometric graph.

  2. Calculate or simplify this relation.

    −1≤cos⁡x≤1⇒−1≤y≤3-1\le\cos x\le1\Rightarrow -1\le y\le3

  3. Both extremes are attained.

[-1,3].

Checks and common pitfalls: Both extremes are attained.

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.

03 / Transfer#Worked example

Is sine increasing on the whole real line? Give a counterexample.

y=sin⁡xy=\sin x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Hint 2
Compare inputs on opposite sides of a peak.
Worked solution

  1. Read amplitude, period and allowed intervals directly from the trigonometric graph.

  2. Calculate or simplify this relation.

    sin⁡(π/2)=1>0=sin⁡π\sin(\pi/2)=1>0=\sin\pi

  3. Monotonicity must be specified on suitable intervals.

No: π/2<π but sin(π/2)>sin π.

Checks and common pitfalls: Monotonicity must be specified on suitable intervals.

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

The least positive period is cπ. Find c.

y=sin⁡(3x)y=\sin(3x)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Hint 2
The input angle must increase by one complete cycle.
Worked solution

  1. Read amplitude, period and allowed intervals directly from the trigonometric graph.

  2. Calculate or simplify this relation.

    3T=2π⇒T=23π3T=2\pi\Rightarrow T=\frac2{3}\pi

  3. Frequency multiplies the input and divides the period.

The requested value is 0.66666667.

Checks and common pitfalls: Frequency multiplies the input and divides the period.

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

05 / Foundation#Your turn

State the full range.

y=3cos⁡x+1y=3\cos x+1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Hint 2
Scale the cosine bounds and shift upward.
Worked solution

  1. Read amplitude, period and allowed intervals directly from the trigonometric graph.

  2. Calculate or simplify this relation.

    −1≤cos⁡x≤1⇒−2≤y≤4-1\le\cos x\le1\Rightarrow -2\le y\le4

  3. Both extremes are attained.

[-2,4].

Checks and common pitfalls: Both extremes are attained.

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.

06 / Foundation#Your turn

State the excluded inputs for tangent.

y=tan⁡xy=\tan x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Hint 2
A tangent denominator vanishes on vertical unit-circle rays.
Worked solution

  1. Read amplitude, period and allowed intervals directly from the trigonometric graph.

  2. Calculate or simplify this relation.

    cos⁡x=0  ⟺  x=π/2+nπ\cos x=0\iff x=\pi/2+n\pi

  3. The domain has infinitely many separated intervals.

x=π/2+nπ, where n is any integer.

Checks and common pitfalls: The domain has infinitely many separated intervals.

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.

07 / Foundation#Your turn

Count the zeros on the closed interval.

y=sin⁡x;0≤x≤3πy=\sin x;\quad0\le x\le3\pi
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Hint 2
Include both interval endpoints if they are zeros.
Worked solution

  1. Read amplitude, period and allowed intervals directly from the trigonometric graph.

  2. Calculate or simplify this relation.

    x=nπ,n=0,1,…,3x=n\pi,\quad n=0,1,\ldots,3

  3. The number of subintervals differs from the number of endpoints.

The requested value is 4.

Checks and common pitfalls: The number of subintervals differs from the number of endpoints.

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

08 / Standard#Your turn

Classify the parity of tangent on its natural domain.

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

Working and explanation

BUILD THE REASONING

Hint 1
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Hint 2
Use sine oddness and cosine evenness.
Worked solution

  1. Read amplitude, period and allowed intervals directly from the trigonometric graph.

  2. Calculate or simplify this relation.

    tan⁡(−x)=−sin⁡xcos⁡x=−tan⁡x\tan(-x)=\frac{-\sin x}{\cos x}=-\tan x

  3. The excluded inputs are symmetric about zero.

Odd.

Checks and common pitfalls: The excluded inputs are symmetric about zero.

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.

09 / Standard#Your turn

Give every x where sine reaches its maximum.

y=sin⁡xy=\sin x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Hint 2
Use the top of the unit circle and full-turn repeats.
Worked solution

  1. Read amplitude, period and allowed intervals directly from the trigonometric graph.

  2. Calculate or simplify this relation.

    sin⁡x=1  ⟺  x=π/2+2nπ\sin x=1\iff x=\pi/2+2n\pi

  3. Half-turn repeats would switch the maximum to the minimum.

x=π/2+2nπ for integer n.

Checks and common pitfalls: Half-turn repeats would switch the maximum to the minimum.

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

Find the minimum of cosine on [0,π] and where it occurs.

y=cos⁡x,0≤x≤πy=\cos x,\quad0\le x\le\pi
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Follow the horizontal coordinate on the upper unit semicircle.
Hint 2
The horizontal coordinate decreases throughout this interval.
Worked solution

  1. Follow the horizontal coordinate on the upper unit semicircle.

  2. Calculate or simplify this relation.

    cos⁡0=1,cos⁡π=−1\cos0=1,\quad\cos\pi=-1

  3. The closed endpoint π is included.

Minimum −1 at x=π.

Checks and common pitfalls: The closed endpoint π is included.

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

The least positive period is cπ. Find c.

y=sin⁡(4x)y=\sin(4x)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Hint 2
The input angle must increase by one complete cycle.
Worked solution

  1. Read amplitude, period and allowed intervals directly from the trigonometric graph.

  2. Calculate or simplify this relation.

    4T=2π⇒T=24π4T=2\pi\Rightarrow T=\frac2{4}\pi

  3. Frequency multiplies the input and divides the period.

The requested value is 0.5.

Checks and common pitfalls: Frequency multiplies the input and divides the period.

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

12 / Transfer#Your turn

State the full range.

y=4cos⁡x+1y=4\cos x+1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Read amplitude, period and allowed intervals directly from the trigonometric graph.
Hint 2
Scale the cosine bounds and shift upward.
Worked solution

  1. Read amplitude, period and allowed intervals directly from the trigonometric graph.

  2. Calculate or simplify this relation.

    −1≤cos⁡x≤1⇒−3≤y≤5-1\le\cos x\le1\Rightarrow -3\le y\le5

  3. Both extremes are attained.

[-3,5].

Checks and common pitfalls: Both extremes are attained.

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.

13 / Transfer#Your turn

Find the least positive period of sin²x, expressed as cπ; enter c and justify why a half-sized candidate fails.

y=sin⁡2xy=\sin^2x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Squaring removes the sign change after π.
Hint 2
Use the zero pattern x=nπ to locate possible periods.
Worked solution

  1. Squaring removes the sign change after π.

  2. Calculate or simplify this relation.

    sin⁡2(x+π)=sin⁡2x;f(0)=0≠1=f(π/2)\sin^2(x+\pi)=\sin^2x;\quad f(0)=0\ne1=f(\pi/2)

  3. Any positive period must carry zero at 0 to another zero, so π is least.

The requested value is 1.

Checks and common pitfalls: Any positive period must carry zero at 0 to another zero, so π is least.

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 graphs and properties of trigonometric functions?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Use graphs and properties of trigonometric functions with explicit angle units and domains.
    • Cycle and range: Sine and cosine repeat every 2π and take values from −1 to 1.
    • Tangent restrictions: Tangent excludes zeros of cosine and has period π.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Sine and cosine repeat every 2π and take values from −1 to 1.
    • Expected reasoning: Tangent excludes zeros of cosine and has period π.
    • Expected correction: Periodicity does not imply global monotonicity.

    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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