LEARN · EXPLAIN · REVISE

Simplify sine after a three-quarter-turn shift.

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#Your turn

\sin(3\pi/2+\alpha)

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the addition formula or a unit-circle rotation.

Hint 2

sin(3π/2)=−1 and cos(3π/2)=0.

Worked solution

Use the addition formula or a unit-circle rotation.
Calculate or simplify this relation.
$\sin(3\pi/2+\alpha)=(-1)\cos\alpha+0\sin\alpha$
Quarter-turn shifts exchange the coordinate roles, with a sign determined by orientation.

−cos α.

Checks and common pitfalls: Quarter-turn shifts exchange the coordinate roles, with a sign determined by orientation.

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.