LEARN · EXPLAIN · REVISE

Derive the cotangent reduction identity and state a domain restriction.

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 / Standard#Your turn

\cot(\pi-\alpha)

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write cotangent as cosine divided by sine.

Hint 2

Apply the two reflection identities separately.

Worked solution

Write cotangent as cosine divided by sine.
Calculate or simplify this relation.
$\cot(\pi-\alpha)=\frac{-\cos\alpha}{\sin\alpha}=-\cot\alpha$
Both sides require a nonzero sine denominator.

−cot α, where sin α≠0.

Checks and common pitfalls: Both sides require a nonzero sine denominator.

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.