LEARN · EXPLAIN · REVISE

Find sin θ from the terminal-ray point.

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

高一必修第一册(A版).pdf · 5.2 · PDF 184 / printed page 177

TOPIC 01

Concept of trigonometric functions

Build understanding of concept of trigonometric functions through definitions, contrasting cases and justified applications.

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

01 / Foundation#Your turn

P=(9,12)

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use coordinates on the terminal ray and a positive radius.

Hint 2

Sine is vertical coordinate divided by radius.

Worked solution

Use coordinates on the terminal ray and a positive radius.
Calculate or simplify this relation.
$r=\sqrt{(9)^2+(12)^2}=15;\quad\sin\theta=4/5$
Scaling a point along the ray does not change the ratio.

The requested value is 0.8.

Checks and common pitfalls: Scaling a point along the ray does not change the ratio.

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

Teacher preparation and assessment

What must be true before using the main rule for concept of trigonometric functions?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Use concept of trigonometric functions with explicit angle units and domains.
Coordinate definitions: Use a positive radius and signed coordinates.
$\sin\theta=y/r,\quad\cos\theta=x/r$
Basic identity: The unit-circle equation gives sine squared plus cosine squared equals one.
$\sin^2\theta+\cos^2\theta=1$
Close with: conditions → representation → reasoning → check.

Expected reasoning: Use a positive radius and signed coordinates.
Expected reasoning: The unit-circle equation gives sine squared plus cosine squared equals one.
Expected correction: Select signs using the quadrant after taking a square root.