LEARN · EXPLAIN · REVISE

The least positive period is cπ. Find c.

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

TOPIC 01

Graphs and properties of trigonometric functions

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

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.

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

01 / Foundation#Worked example

y=\sin(2x)

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

Read amplitude, period and allowed intervals directly from the trigonometric graph.
Calculate or simplify this relation.
$2T=2\pi\Rightarrow T=\frac2{2}\pi$
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.

Teacher preparation and assessment

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?

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.

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.