Kenneth ChengMathematical modeling
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LEARN · EXPLAIN · REVISE

General angles and radian measure

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

高一必修 第一册(A版).pdf · 5.1 · PDF 175 / printed page 168

Revisit first: Basic properties of functions

TOPIC 01

General angles and radian measure

Build understanding of general angles and radian measure through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use general angles and radian measure 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.

Directed angles

Positive rotation is anticlockwise; full turns preserve the terminal ray.

Radian definition

Radian measure is arc length divided by radius.

θ=s/r\theta=s/r

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 general angles and radian measure 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

Write the angle as cπ radians; find c.

60∘60^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
One straight angle is π radians.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    60∘⋅π180∘=26π60^{\circ}\cdot\frac{\pi}{180^{\circ}}=\frac{2}6\pi

  3. The numerical coefficient changes with the unit, but the geometric angle does not.

The requested value is 0.33333333.

Checks and common pitfalls: The numerical coefficient changes with the unit, but the geometric angle does not.

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

02 / Standard#Worked example

Convert to degrees.

2π4\frac{2\pi}{4}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
Multiply the radian measure by 180/π.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    2π4⋅180∘π=90∘\frac{2\pi}4\cdot\frac{180^{\circ}}{\pi}=90^{\circ}

  3. State degrees in the result.

The requested value is 90.

Checks and common pitfalls: State degrees in the result.

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

03 / Transfer#Worked example

How many coterminal representatives of 30° lie in this closed interval?

[−720∘,720∘][-720^{\circ},720^{\circ}]
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
Write every coterminal angle as 30°+360°n.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    −720≤30+360n≤720⇒−2≤n≤1-720\le30+360n\le720\Rightarrow -2\le n\le1

  3. Integer n must satisfy both bounds.

The requested value is 4.

Checks and common pitfalls: Integer n must satisfy both bounds.

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

04 / Foundation#Your turn

Write the angle as cπ radians; find c.

90∘90^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
One straight angle is π radians.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    90∘⋅π180∘=36π90^{\circ}\cdot\frac{\pi}{180^{\circ}}=\frac{3}6\pi

  3. The numerical coefficient changes with the unit, but the geometric angle does not.

The requested value is 0.5.

Checks and common pitfalls: The numerical coefficient changes with the unit, but the geometric angle does not.

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

05 / Foundation#Your turn

Convert to degrees.

3π4\frac{3\pi}{4}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
Multiply the radian measure by 180/π.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    3π4⋅180∘π=135∘\frac{3\pi}4\cdot\frac{180^{\circ}}{\pi}=135^{\circ}

  3. State degrees in the result.

The requested value is 135.

Checks and common pitfalls: State degrees in the result.

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

06 / Foundation#Your turn

A circle has radius r and angle θ radians. Find the arc length for these values.

r=3,θ=2r=3,\quad\theta=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
The formula s=rθ requires radians.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    s=rθ=3⋅2=6s=r\theta=3\cdot2=6

  3. Arc length has the same length unit as the radius.

The requested value is 6.

Checks and common pitfalls: Arc length has the same length unit as the radius.

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

07 / Foundation#Your turn

Find the sector area for the stated radian angle.

r=3,θ=1r=3,\quad\theta=1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
A sector takes θ/(2π) of the circle area.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    S=12r2θ=92S=\frac12r^2\theta=\frac{9}2

  3. The area unit is the square of the radius unit.

The requested value is 4.5.

Checks and common pitfalls: The area unit is the square of the radius unit.

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

08 / Standard#Your turn

Give the coterminal angle in [0°,360°).

1125∘1125^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
Subtract complete turns.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    1125∘−3⋅360∘=45∘1125^{\circ}-3\cdot360^{\circ}=45^{\circ}

  3. Coterminal angles can differ as real numbers while sharing a terminal ray.

The requested value is 45.

Checks and common pitfalls: Coterminal angles can differ as real numbers while sharing a terminal ray.

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

09 / Standard#Your turn

Which quadrant contains the terminal ray? Explain.

−150∘-150^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
Replace the angle by a coterminal angle between zero and one turn.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    −150∘+360∘=210∘-150^{\circ}+360^{\circ}=210^{\circ}

  3. A negative angle indicates clockwise rotation, not a negative quadrant.

Third quadrant.

Checks and common pitfalls: A negative angle indicates clockwise rotation, not a negative quadrant.

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

How many coterminal representatives of 30° lie in this closed interval?

[−1080∘,1080∘][-1080^{\circ},1080^{\circ}]
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
Write every coterminal angle as 30°+360°n.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    −1080≤30+360n≤1080⇒−3≤n≤2-1080\le30+360n\le1080\Rightarrow -3\le n\le2

  3. Integer n must satisfy both bounds.

The requested value is 6.

Checks and common pitfalls: Integer n must satisfy both bounds.

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

11 / Standard#Your turn

Write the angle as cπ radians; find c.

120∘120^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
One straight angle is π radians.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    120∘⋅π180∘=46π120^{\circ}\cdot\frac{\pi}{180^{\circ}}=\frac{4}6\pi

  3. The numerical coefficient changes with the unit, but the geometric angle does not.

The requested value is 0.66666667.

Checks and common pitfalls: The numerical coefficient changes with the unit, but the geometric angle does not.

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

12 / Transfer#Your turn

Convert to degrees.

4π4\frac{4\pi}{4}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
Multiply the radian measure by 180/π.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    4π4⋅180∘π=180∘\frac{4\pi}4\cdot\frac{180^{\circ}}{\pi}=180^{\circ}

  3. State degrees in the result.

The requested value is 180.

Checks and common pitfalls: State degrees in the result.

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

13 / Transfer#Your turn

A circle has radius r and angle θ radians. Find the arc length for these values.

r=4,θ=2r=4,\quad\theta=2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep degrees and radians distinct and use the definition of angular measure.
Hint 2
The formula s=rθ requires radians.
Worked solution

  1. Keep degrees and radians distinct and use the definition of angular measure.

  2. Calculate or simplify this relation.

    s=rθ=4⋅2=8s=r\theta=4\cdot2=8

  3. Arc length has the same length unit as the radius.

The requested value is 8.

Checks and common pitfalls: Arc length has the same length unit as the radius.

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 general angles and radian measure?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Use general angles and radian measure with explicit angle units and domains.
    • Directed angles: Positive rotation is anticlockwise; full turns preserve the terminal ray.
    • Radian definition: Radian measure is arc length divided by radius.
      θ=s/r\theta=s/r
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Positive rotation is anticlockwise; full turns preserve the terminal ray.
    • Expected reasoning: Radian measure is arc length divided by radius.
    • Expected correction: The arc and sector formulas require radians.

    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 ↗