LEARN · EXPLAIN · REVISE

Disprove the claim that cos(π−α)=cos α for every α.

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

高一必修第一册(A版).pdf · 5.3 · PDF 195 / printed page 188

TOPIC 01

Reduction formulas

Build understanding of reduction formulas through definitions, contrasting cases and justified applications.

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

01 / Transfer#Worked example

\cos(\pi-\alpha)=\cos\alpha

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use unit-circle symmetry and periodicity; determine the sign before simplifying.

Hint 2

Choose an angle with nonzero cosine.

Worked solution

Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Calculate or simplify this relation.
$\cos\pi=-1,\quad\cos0=1$
The correct identity has a minus sign.

Take α=0: −1≠1.

Checks and common pitfalls: The correct identity has a minus sign.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

Use reduction formulas with explicit angle units and domains.
Symmetry: Reflections determine which coordinate changes sign.
Periodicity: A full turn preserves sine and cosine; a half turn preserves tangent when defined.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Reflections determine which coordinate changes sign.
Expected reasoning: A full turn preserves sine and cosine; a half turn preserves tangent when defined.
Expected correction: A reflected cosine often changes sign.