LEARN · EXPLAIN · REVISE

Is sine increasing on the whole real line? Give a counterexample.

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#Worked example

y=\sin x

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

Compare inputs on opposite sides of a peak.

Worked solution

Read amplitude, period and allowed intervals directly from the trigonometric graph.
Calculate or simplify this relation.
$\sin(\pi/2)=1>0=\sin\pi$
Monotonicity must be specified on suitable intervals.

No: π/2<π but sin(π/2)>sin π.

Checks and common pitfalls: Monotonicity must be specified on suitable intervals.

Reasoning checklist · self / teacher assessment

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.