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Unit-circle coordinates and signs

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

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.

Coordinates before ratios

Coordinate signs survive normalisation; length alone does not determine a quadrant.

For a nonzero terminal-ray point, divide both coordinates by its positive radius.

\cos\theta=x/r,\quad\sin\theta=y/r,\quad r=\sqrt{x^2+y^2}

Scale the point by its radius.
$(x,y)\mapsto(x/r,y/r)$
Pythagoras becomes a unit-circle constraint.
$\cos^2\theta+\sin^2\theta=1$

When it applies

r>0; the origin has no terminal direction.

Counterexample: P=(−3,4) has negative cosine even though every side length is positive.

Link an oriented ray, signed coordinates and a unit-circle point.

Direction and admissible ratios

A rotation model needs orientation and units before a trig value is chosen.

Counterclockwise angles are positive; tangent is undefined when the x-coordinate is zero.

\tan\theta=\sin\theta/\cos\theta

Add complete turns without changing the terminal point.
$\theta\sim\theta+360^\circ k$
Inspect the divisor before using tangent.
$\cos\theta\ne0$

When it applies

Degree and radian measures must not be mixed; tangent requires nonzero cosine.

Counterexample: tan 90° is undefined, not zero.

Compare degree/radian labels and clockwise/counterclockwise arrows.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Which coordinate changes when θ is reflected to −θ or π−θ?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Physical point (-1,1.73205); divide by positive r=2 for the unit-circle coordinates. Positive angles turn counterclockwise.

Numerical data for the diagram; printable
Quantity	Value
θ (rad)	2.0944
x	-1
y	1.73205
cos θ	-0.5
sin θ	0.86603
tan θ	-1.73205

Explain: Track signed coordinates and the tangent denominator while varying radius and angle.

Transfer: Choose two different-radius points with identical trig ratios.

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 / Foundation#Worked example

The terminal ray passes through P. Find cosine.

P=(-3,4)

P is nonzero.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Find the positive radius.

Hint 2

Cosine uses the horizontal coordinate.

Worked solution

Normalise by the distance from the origin.
$r=\sqrt{9+16}=5$
Retain the sign of x.
$\cos\theta=-3/5$

−3/5.

Checks and common pitfalls: Distance is positive, while the leftward coordinate is negative.

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

02 / Standard#Worked example

Find cosine from the given sine and quadrant.

\sin\theta=3/5,\quad\theta\text{ in quadrant II}

First step provided — complete the reasoning:Find the squared cosine.

\cos^2\theta=1-9/25=16/25

Angle lies strictly in quadrant II.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Use the unit-circle identity.

Hint 2

Choose the negative square-root branch in quadrant II.

Worked solution

Find the squared cosine.
$\cos^2\theta=1-9/25=16/25$
Apply the quadrant sign.
$\cos\theta=-4/5$

−4/5.

Checks and common pitfalls: The identity alone gives two signs; the quadrant selects one.

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

03 / 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.

04 / Foundation#Your turn

Evaluate.

\sin150^\circ

Angles are in degrees.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

The reference angle is 30°.

Hint 2

Sine is positive in quadrant II.

Worked solution

Reflect a 30° point across the y-axis.
$150^\circ=180^\circ-30^\circ$
The vertical coordinate is preserved.
$\sin150^\circ=1/2$

1/2.

Checks and common pitfalls: Reflection changes cosine, not this sine.

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

05 / Foundation#Your turn

Evaluate.

\cos240^\circ

Angles are in degrees.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

240° is in quadrant III.

Hint 2

Its reference angle is 60°.

Worked solution

Separate the half-turn.
$240^\circ=180^\circ+60^\circ$
The horizontal coordinate reverses sign.
$\cos240^\circ=-1/2$

−1/2.

Checks and common pitfalls: A half-turn reverses both coordinates.

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

06 / Foundation#Your turn

Explain the status of tan 90°.

\tan90^\circ

Real trigonometric functions are used.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Use sin/cos.

Hint 2

The denominator is zero.

Worked solution

Read the point at a quarter-turn.
$(\cos90^\circ,\sin90^\circ)=(0,1)$
The quotient cannot be a real number.
$1/0\text{ undefined}$

Undefined.

Checks and common pitfalls: It is neither zero nor a real infinite value.

Reasoning checklist · self / teacher assessment

Read the point at a quarter-turn.
The quotient cannot be a real number.

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

07 / Foundation#Your turn

Give the unit-circle coordinates.

\theta=-\pi/3

The input is in radians.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

A negative angle rotates clockwise.

Hint 2

The magnitude is 60°.

Worked solution

Locate quadrant IV.
$-\pi/3=-60^\circ$
Cosine is positive and sine negative.
$P=(1/2,-\sqrt3/2)$

(1/2,−√3/2).

Checks and common pitfalls: The radian input must be interpreted as an angle, not a degree number.

Reasoning checklist · self / teacher assessment

Locate quadrant IV.
Cosine is positive and sine negative.

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

08 / Standard#Your turn

Find sine.

\cos\theta=-12/13,\quad\theta\text{ in quadrant III}

Angle is in quadrant III.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Use squared coordinates summing to 1.

Hint 2

Sine is negative in quadrant III.

Worked solution

Find the squared vertical coordinate.
$\sin^2\theta=1-144/169=25/169$
Select the negative branch.
$\sin\theta=-5/13$

−5/13.

Checks and common pitfalls: The sign is supplied by the quadrant, not by the square root computation.

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

09 / Standard#Your turn

Find cosine exactly.

\tan\theta=-2,\quad\theta\text{ in quadrant IV}

Cosine is nonzero; quadrant IV fixes its sign.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Use a ray with coordinate ratio −2.

Hint 2

Choose point (1,−2).

Worked solution

Choose a representative point with the correct quadrant.
$P=(1,-2),\quad r=\sqrt5$
Normalise the horizontal coordinate.
$\cos\theta=1/\sqrt5$

1/√5.

Checks and common pitfalls: A ratio does not fix length, but normalisation removes that freedom.

Reasoning checklist · self / teacher assessment

Choose a representative point with the correct quadrant.
Normalise the horizontal coordinate.

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

10 / Standard#Your turn

Give the angle in degrees in [0,360).

P=(0,-2)

Answer in degrees, 0≤θ<360°.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Locate the negative y-axis.

Hint 2

Use the positive counterclockwise representative.

Worked solution

The terminal direction is downward.
$(x/r,y/r)=(0,-1)$
Choose the requested interval representative.
$\theta=270^\circ$

270°.

Checks and common pitfalls: −90° is coterminal but outside the requested interval.

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

11 / Standard#Your turn

Explain geometrically why sin(π−α)=sin α.

0<\alpha<\pi/2

α is acute and measured in radians.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Reflect across the y-axis.

Hint 2

Reflection preserves the vertical coordinate.

Worked solution

Write the original unit-circle point.
$P=(\cos\alpha,\sin\alpha)$
Reflect to angle π−α.
$P_{\mathrm{reflected}}=(-\cos\alpha,\sin\alpha)$

The equal y-coordinates prove the identity.

Checks and common pitfalls: This explains which coordinate changes sign instead of memorising a formula.

Reasoning checklist · self / teacher assessment

Write the original unit-circle point.
Reflect to angle π−α.

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

12 / Transfer#Your turn

A point starts at (2,0) and rotates clockwise by 120° about the origin. Find its final height and position; explain the sign of the height.

r=2,\quad\theta=-120^\circ

Rotation about the origin; positive y is upward.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Height is a signed vertical coordinate, not distance from the x-axis.

Hint 2

Use radius times the unit-circle coordinates.

Worked solution

Determine the signed trigonometric values.
$\cos(-120^\circ)=-1/2,\quad\sin(-120^\circ)=-\sqrt3/2$
Scale by the physical radius.
$P=(-1,-\sqrt3)$

Height −√3 and position (−1,−√3).

Checks and common pitfalls: The point lies below the axis after this clockwise turn; distance below it would be positive √3.

Reasoning checklist · self / teacher assessment

Determine the signed trigonometric values.
Scale by the physical radius.

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

13 / Transfer#Your turn

Two exact readings claim sine=0.8 and cosine=0.8 for one angle. Decide whether any real angle fits and explain whether changing quadrant could repair them.

s=0.8,\quad c=0.8

Readings are stated as exact, not rounded measurements.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Check the unit-circle norm before recovering an angle.

Hint 2

The squared readings sum to 1.28.

Worked solution

Test the necessary identity.
$s^2+c^2=0.64+0.64=1.28\ne1$
Sign changes do not alter squared length.
$(\pm0.8)^2+(\pm0.8)^2=1.28$

No angle fits; changing quadrant cannot repair exact readings.

Checks and common pitfalls: This is a consistency test rather than an inverse-trig calculation.

Reasoning checklist · self / teacher assessment

Test the necessary identity.
Sign changes do not alter squared length.

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

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

Choose a foundation skill to revisit ↗

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.