Kenneth ChengMathematical modeling
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Concept of trigonometric functions

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

高一必修 第一册(A版).pdf · 5.2 · PDF 184 / printed page 177

Revisit first: General angles and radian measure

TOPIC 01

Concept of trigonometric functions

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

What you will be able to explain

  • Use concept 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.

Coordinate definitions

Use a positive radius and signed coordinates.

sin⁡θ=y/r,cos⁡θ=x/r\sin\theta=y/r,\quad\cos\theta=x/r

Basic identity

The unit-circle equation gives sine squared plus cosine squared equals one.

sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1

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 concept 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

Find sin θ from the terminal-ray point.

P=(6,8)P=(6,8)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use coordinates on the terminal ray and a positive radius.
Hint 2
Sine is vertical coordinate divided by radius.
Worked solution

  1. Use coordinates on the terminal ray and a positive radius.

  2. Calculate or simplify this relation.

    r=(6)2+(8)2=10;sin⁡θ=4/5r=\sqrt{(6)^2+(8)^2}=10;\quad\sin\theta=4/5

  3. Scaling a point along the ray does not change the ratio.

The requested value is 0.8.

Checks and common pitfalls: Scaling a point along the ray does not change the ratio.

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

02 / Standard#Worked example

Find cos θ when θ is in quadrant II.

sin⁡θ=3/5\sin\theta=3/5
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use coordinates on the terminal ray and a positive radius.
Hint 2
Use the identity and then select the sign.
Worked solution

  1. Use coordinates on the terminal ray and a positive radius.

  2. Calculate or simplify this relation.

    cos⁡2θ=1−9/25=16/25;cos⁡θ=−4/5\cos^2\theta=1-9/25=16/25;\quad\cos\theta=-4/5

  3. The quadrant determines the negative cosine.

The requested value is -0.8.

Checks and common pitfalls: The quadrant determines the negative cosine.

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

03 / Transfer#Worked example

Find the radius of this terminal-ray point.

P=(−10,24)P=(-10,24)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use coordinates on the terminal ray and a positive radius.
Hint 2
Use the distance formula, not the signed coordinate sum.
Worked solution

  1. Use coordinates on the terminal ray and a positive radius.

  2. Calculate or simplify this relation.

    r=100+576=26r=\sqrt{100+576}=26

  3. The radius is positive regardless of quadrant.

The requested value is 26.

Checks and common pitfalls: The radius is positive regardless of quadrant.

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

04 / Foundation#Your turn

Find sin θ from the terminal-ray point.

P=(9,12)P=(9,12)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use coordinates on the terminal ray and a positive radius.
Hint 2
Sine is vertical coordinate divided by radius.
Worked solution

  1. Use coordinates on the terminal ray and a positive radius.

  2. Calculate or simplify this relation.

    r=(9)2+(12)2=15;sin⁡θ=4/5r=\sqrt{(9)^2+(12)^2}=15;\quad\sin\theta=4/5

  3. Scaling a point along the ray does not change the ratio.

The requested value is 0.8.

Checks and common pitfalls: Scaling a point along the ray does not change the ratio.

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

05 / Foundation#Your turn

Find cos θ when θ is in quadrant III.

sin⁡θ=−12/13\sin\theta=-12/13
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the Pythagorean identity and the quadrant.
Hint 2
Cosine is negative in quadrant III.
Worked solution

  1. Use the Pythagorean identity and the quadrant.

  2. Calculate or simplify this relation.

    cos⁡2θ=1−144/169=25/169;cos⁡θ=−5/13\cos^2\theta=1-144/169=25/169;\quad\cos\theta=-5/13

  3. The square-root sign comes from the quadrant, not from the sign of sine alone.

The requested value is -0.38461538.

Checks and common pitfalls: The square-root sign comes from the quadrant, not from the sign of sine alone.

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

06 / Foundation#Your turn

Find tan θ from the terminal-ray point.

P=(12,−9)P=(12,-9)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use coordinates on the terminal ray and a positive radius.
Hint 2
Tangent uses y divided by x.
Worked solution

  1. Use coordinates on the terminal ray and a positive radius.

  2. Calculate or simplify this relation.

    tan⁡θ=y/x=−3/4\tan\theta=y/x=-3/4

  3. The denominator must be nonzero.

The requested value is -0.75.

Checks and common pitfalls: The denominator must be nonzero.

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

07 / Foundation#Your turn

Explain why tangent is undefined at 90°.

tan⁡θ=sin⁡θ/cos⁡θ\tan\theta=\sin\theta/\cos\theta
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use coordinates on the terminal ray and a positive radius.
Hint 2
Locate the point at the top of the unit circle.
Worked solution

  1. Use coordinates on the terminal ray and a positive radius.

  2. Calculate or simplify this relation.

    sin⁡90∘=1,cos⁡90∘=0\sin90^{\circ}=1,\quad\cos90^{\circ}=0

  3. Undefined is different from an extremely large real value.

cos 90° is zero, so the ratio divides by zero.

Checks and common pitfalls: Undefined is different from an extremely large real value.

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.

08 / Standard#Your turn

Determine whether the two claimed trigonometric values can occur together.

sin⁡θ=4/5,cos⁡θ=4/5\sin\theta=4/5,\quad\cos\theta=4/5
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use coordinates on the terminal ray and a positive radius.
Hint 2
Apply the unit-circle equation.
Worked solution

  1. Use coordinates on the terminal ray and a positive radius.

  2. Calculate or simplify this relation.

    (4/5)2+(4/5)2=32/25≠1(4/5)^2+(4/5)^2=32/25\ne1

  3. Each value being individually between −1 and 1 is insufficient.

No, because their squared sum is not one.

Checks and common pitfalls: Each value being individually between −1 and 1 is insufficient.

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

In which quadrant is the angle, given both signs?

sin⁡θ<0,cos⁡θ>0\sin\theta<0,\quad\cos\theta>0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use coordinates on the terminal ray and a positive radius.
Hint 2
Interpret sine and cosine as signed coordinates.
Worked solution

  1. Use coordinates on the terminal ray and a positive radius.

  2. Calculate or simplify this relation.

    y<0,x>0y<0,\quad x>0

  3. Angles on the axes cannot satisfy both strict inequalities.

Fourth quadrant.

Checks and common pitfalls: Angles on the axes cannot satisfy both strict inequalities.

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 radius of this terminal-ray point.

P=(−15,36)P=(-15,36)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use coordinates on the terminal ray and a positive radius.
Hint 2
Use the distance formula, not the signed coordinate sum.
Worked solution

  1. Use coordinates on the terminal ray and a positive radius.

  2. Calculate or simplify this relation.

    r=225+1296=39r=\sqrt{225+1296}=39

  3. The radius is positive regardless of quadrant.

The requested value is 39.

Checks and common pitfalls: The radius is positive regardless of quadrant.

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

11 / Standard#Your turn

Find sin θ from the terminal-ray point.

P=(12,16)P=(12,16)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use coordinates on the terminal ray and a positive radius.
Hint 2
Sine is vertical coordinate divided by radius.
Worked solution

  1. Use coordinates on the terminal ray and a positive radius.

  2. Calculate or simplify this relation.

    r=(12)2+(16)2=20;sin⁡θ=4/5r=\sqrt{(12)^2+(16)^2}=20;\quad\sin\theta=4/5

  3. Scaling a point along the ray does not change the ratio.

The requested value is 0.8.

Checks and common pitfalls: Scaling a point along the ray does not change the ratio.

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

12 / Transfer#Your turn

Can an angle in quadrant IV have sine 4/5? Explain.

θ in quadrant IV,sin⁡θ=4/5\theta\text{ in quadrant IV},\quad\sin\theta=4/5
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the signed unit-circle coordinate.
Hint 2
The vertical coordinate is negative in quadrant IV.
Worked solution

  1. Use the signed unit-circle coordinate.

  2. Calculate or simplify this relation.

    y<0, r>0⇒sin⁡θ=y/r<0y<0,\ r>0\Rightarrow\sin\theta=y/r<0

  3. The given assumptions contradict each other before any further calculation.

No.

Checks and common pitfalls: The given assumptions contradict each other before any further calculation.

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 tan θ from the terminal-ray point.

P=(16,−12)P=(16,-12)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use coordinates on the terminal ray and a positive radius.
Hint 2
Tangent uses y divided by x.
Worked solution

  1. Use coordinates on the terminal ray and a positive radius.

  2. Calculate or simplify this relation.

    tan⁡θ=y/x=−3/4\tan\theta=y/x=-3/4

  3. The denominator must be nonzero.

The requested value is -0.75.

Checks and common pitfalls: The denominator must be nonzero.

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

    Board plan

    • Use concept of trigonometric functions with explicit angle units and domains.
    • Coordinate definitions: Use a positive radius and signed coordinates.
      sin⁡θ=y/r,cos⁡θ=x/r\sin\theta=y/r,\quad\cos\theta=x/r
    • Basic identity: The unit-circle equation gives sine squared plus cosine squared equals one.
      sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Use a positive radius and signed coordinates.
    • Expected reasoning: The unit-circle equation gives sine squared plus cosine squared equals one.
    • Expected correction: Select signs using the quadrant after taking a square root.

    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 ↗