LEARN · EXPLAIN · REVISE

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

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 / Transfer#Your turn

y=\sin^2x

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

Squaring removes the sign change after π.
Calculate or simplify this relation.
$\sin^2(x+\pi)=\sin^2x;\quad f(0)=0\ne1=f(\pi/2)$
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.

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.