LEARN · EXPLAIN · REVISE

Write the angle as cπ radians; find c.

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

TOPIC 01

General angles and radian measure

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

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

01 / Foundation#Worked example

60^{\circ}

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

Keep degrees and radians distinct and use the definition of angular measure.
Calculate or simplify this relation.
$60^{\circ}\cdot\frac{\pi}{180^{\circ}}=\frac{2}6\pi$
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.

Teacher preparation and assessment

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?

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.
$\theta=s/r$
Close with: conditions → representation → reasoning → check.

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.