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Which sign pair is correct in quadrant III?

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

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

TOPIC 01

Unit-circle coordinates and signs

Read sine and cosine as coordinates, retaining angle direction and quadrant information.

What you will be able to explain

  • Normalise a point to the unit circle and explain coordinate signs.
  • Use a norm check to judge whether proposed sine/cosine data are possible.

Try it. Leave your reasoning visible.

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

01 / Foundation#Your turn

Which sign pair is correct in quadrant III?

(sin⁡θ,cos⁡θ)(\sin\theta,\cos\theta)
  • Angle lies strictly in quadrant III.
Skills and prerequisite lessons

Scope: JM01 / JM02

  1. (+,+)
  2. (+,−)
  3. (−,+)
  4. (−,−)
Working and explanation
BUILD THE REASONING
Hint 1
Think y then x.
Hint 2
Both coordinates are negative.
Worked solution
  1. Normalisation divides by positive r.
    r>0r>0
  2. Both signs remain negative.
    (sin⁡θ,cos⁡θ)=(−,−)(\sin\theta,\cos\theta)=(-,-)
D: (−,−).
Checks and common pitfalls: Sine and cosine are signed coordinates, not positive lengths.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5Signs and unitsUse diagnostics; require a marked unit-circle point.
8Normalise a pointDerive cosine and sine from P=(−3,4), keeping signs.
10Choose a branchPairs recover a missing coordinate from an identity and quadrant.
12Model and rejectGroups compare clockwise height and impossible sensor readings.
5Exit explanationAsk students to distinguish height from distance below the axis.
Question sequence
  • Which coordinate does sine use?
  • What information chooses the sign after taking a square root?
Board development
  • Normalise by a positive radius.
    (x,y)/r=(cos⁡θ,sin⁡θ)(x,y)/r=(\cos\theta,\sin\theta)
  • Check data before solving for an angle.
    s2+c2=1s^2+c^2=1
Expected correct and incorrect responses
  • “Radius is positive, x is negative, so cosine is negative” is correct reasoning.
  • “Both readings are under 1, so they work” misses their joint constraint.
Follow-up questions
  • Can a sign change repair an incorrect squared norm?
Remediation
  • For sign errors, draw axes and project the point before recalling angle formulas.
Exit check
  • For P=(5,−12), give sine and cosine with a sign reason. Expected −12/13 and 5/13.
Assessment checklist and sample responses
  • 2 marks: coordinate normalisation.
    Full: divide signed coordinates by positive 13; partial: use unsigned lengths only.
  • 2 marks: model interpretation or consistency.
    Full: squared norm 1.28 rejects data in every quadrant; partial: says impossible without a test.
No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.
Curriculum and source notes ↗