LEARN · EXPLAIN · REVISE

A sensor reports a counterclockwise turn of 765° from the positive x-axis on a unit circle. Give its final coordinates and explain why reducing turns is valid.

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

Return to the lesson / paper ↗

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.

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Transfer#Worked example

A sensor reports a counterclockwise turn of 765° from the positive x-axis on a unit circle. Give its final coordinates and explain why reducing turns is valid.

\theta=765^\circ

Radius 1; positive rotation is counterclockwise.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Remove complete turns, not a half-turn.

Hint 2

765−720=45.

Worked solution

Find the coterminal angle.
$765^\circ=2(360^\circ)+45^\circ$
Read the equal positive coordinates at 45°.
$P=(\sqrt2/2,\sqrt2/2)$

(√2/2,√2/2).

Checks and common pitfalls: Full turns preserve the endpoint, while the recorded travel still totals 765°.

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

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Signs and units	Use diagnostics; require a marked unit-circle point.
8	Normalise a point	Derive cosine and sine from P=(−3,4), keeping signs.
10	Choose a branch	Pairs recover a missing coordinate from an identity and quadrant.
12	Model and reject	Groups compare clockwise height and impossible sensor readings.
5	Exit explanation	Ask 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.
$(x,y)/r=(\cos\theta,\sin\theta)$
Check data before solving for an angle.
$s^2+c^2=1$

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.