LEARN · EXPLAIN · REVISE

Find cos θ when θ is in quadrant III.

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

\sin\theta=-12/13

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the Pythagorean identity and the quadrant.

Hint 2

Cosine is negative in quadrant III.

Worked solution

Use the Pythagorean identity and the quadrant.
Calculate or simplify this relation.
$\cos^2\theta=1-144/169=25/169;\quad\cos\theta=-5/13$
The square-root sign comes from the quadrant, not from the sign of sine alone.

The requested value is -0.38461538.

Checks and common pitfalls: The square-root sign comes from the quadrant, not from the sign of sine alone.

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.