LEARN · EXPLAIN · REVISE

For π<α<3π/2 and cos α=−3/5, determine the sign and value of sin(α/2).

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

高一必修第一册(A版).pdf · 5.5 · PDF 222 / printed page 215

TOPIC 01

Trigonometric identities and transformations

Build understanding of trigonometric identities and transformations through definitions, contrasting cases and justified applications.

Use trigonometric identities and transformations 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 / Standard#Your turn

\pi<\alpha<3\pi/2,\quad\cos\alpha=-3/5;\quad\sin^2(\alpha/2)=(1-\cos\alpha)/2

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Select an identity that matches the structure, retain signs and check denominators.

Hint 2

Locate the half-angle before choosing a square-root sign.

Worked solution

Select an identity that matches the structure, retain signs and check denominators.
Calculate or simplify this relation.
$\pi/2<\alpha/2<3\pi/4;\quad\sin(\alpha/2)=\sqrt{4/5}$
The interval, not merely the terminal-ray quadrant, fixes the half-angle sign; adding 2π to α reverses that sign.

Positive, equal to 2/√5.

Checks and common pitfalls: The interval, not merely the terminal-ray quadrant, fixes the half-angle sign; adding 2π to α reverses that sign.

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 trigonometric identities and transformations?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Use trigonometric identities and transformations with explicit angle units and domains.
Angle sums: Use paired sine and cosine products in the addition identities.
$\sin(a+b)=\sin a\cos b+\cos a\sin b$
Equivalent forms: Choose a double-angle or half-angle form matching the given data.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Use paired sine and cosine products in the addition identities.
Expected reasoning: Choose a double-angle or half-angle form matching the given data.
Expected correction: Cancelled factors cannot restore originally forbidden inputs.