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Polar coordinates

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

2027 JM02 考試大綱 · 4. Coordinate geometry: polar coordinates · PDF 2 / printed page 2

TOPIC 01

Polar coordinates

Convert representations and eliminate polar variables without losing the geometric range.

What you will be able to explain

Convert representations and eliminate polar variables without losing the geometric range.
Justify the method and check the conditions in a new situation.

Model or definition

Convert representations and eliminate polar variables without losing the geometric range.

x=r\cos\theta,\quad y=r\sin\theta

Conditions

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Does adding π to an angle preserve the point when r stays positive?

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

(r,θ)=(2,45°) gives (1.4142,1.4142). (−r,θ+180°) describes the same point; at r=0 the angle is not unique.

Explain: Compare two admissible cases and explain their different results using the stated model.

Transfer: Compare θ+2π and the signed-radius pair (−r,θ+π).

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find x for the polar point.

(r,\theta)=(2,0)

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use this intermediate relation.

x=r\cos\theta

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$x=2\cos0=2$
Angle zero points along the positive x-axis.

The requested value is 2.

Checks and common pitfalls: Angle zero points along the positive x-axis.

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

02 / Standard#Worked example

Convert the polar curve to Cartesian form.

r=6\cos\theta

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Multiply by r and use r²=x²+y².

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$r^2=2(3)r\cos\theta$
Apply the stated relation and retain its conditions.
$x^2+y^2=6x$
Apply the stated relation and retain its conditions.
$(x-3)^2+y^2=9$
Under r≥0, only cosθ≥0 contributes nonzero points, but the complete Cartesian circle is still traced.

The requested relation or conclusion is shown below.

(x-3)^2+y^2=9

Checks and common pitfalls: Under r≥0, only cosθ≥0 contributes nonzero points, but the complete Cartesian circle is still traced.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

03 / Transfer#Worked example

Describe the displayed polar locus with r≥0, and compare it with its full supporting line.

\theta=\pi/6,\quad r\ge0

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use both coordinate relations and retain the nonnegative radius.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$y=\tan(\pi/6)x, x\ge0$
The polar locus is a ray; dropping the sign restriction would add the opposite ray.

The requested relation or conclusion is shown below.

y=\tan(\pi/6)x,\quad x\ge0

Checks and common pitfalls: The polar locus is a ray; dropping the sign restriction would add the opposite ray.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

04 / Foundation#Your turn

Find x for the polar point.

(r,\theta)=(5,0)

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use this intermediate relation.

x=r\cos\theta

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$x=5\cos0=5$
Angle zero points along the positive x-axis.

The requested value is 5.

Checks and common pitfalls: Angle zero points along the positive x-axis.

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

05 / Foundation#Your turn

Find y for the polar point.

(r,\theta)=(6,\pi/2)

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use this intermediate relation.

y=r\sin\theta

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$y=6\sin(\pi/2)=6$
The π/2 direction lies on the positive y-axis.

The requested value is 6.

Checks and common pitfalls: The π/2 direction lies on the positive y-axis.

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

06 / Foundation#Your turn

Find r≥0 for the Cartesian point.

P=(21,28)

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use this intermediate relation.

r=\sqrt{x^2+y^2}

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$r=\sqrt{441+784}=35$
Use the nonnegative distance from the origin.

The requested value is 35.

Checks and common pitfalls: Use the nonnegative distance from the origin.

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

07 / Foundation#Your turn

Give a positive-radius polar representation of (−t,0).

t=8

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

The point is on the negative x-axis.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$x=8\cos\pi=-8,\quad y=0$
The angle is π rather than zero when radius is positive.

The requested relation or conclusion is shown below.

(r,\theta)=(8,\pi)

Checks and common pitfalls: The angle is π rather than zero when radius is positive.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

08 / Standard#Your turn

Give a positive-radius polar representation of (−t,0).

t=9

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

The point is on the negative x-axis.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$x=9\cos\pi=-9,\quad y=0$
The angle is π rather than zero when radius is positive.

The requested relation or conclusion is shown below.

(r,\theta)=(9,\pi)

Checks and common pitfalls: The angle is π rather than zero when radius is positive.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

09 / Standard#Your turn

Convert the polar curve to Cartesian form.

r=20\cos\theta

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Multiply by r and use r²=x²+y².

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$r^2=2(10)r\cos\theta$
Apply the stated relation and retain its conditions.
$x^2+y^2=20x$
Apply the stated relation and retain its conditions.
$(x-10)^2+y^2=100$
Under r≥0, only cosθ≥0 contributes nonzero points, but the complete Cartesian circle is still traced.

The requested relation or conclusion is shown below.

(x-10)^2+y^2=100

Checks and common pitfalls: Under r≥0, only cosθ≥0 contributes nonzero points, but the complete Cartesian circle is still traced.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

10 / Standard#Your turn

Under the signed-radius convention, show that the two polar pairs represent the same point.

(11,\pi/3),\quad(-11,4\pi/3)

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use this intermediate relation.

cos(θ+π)=−cosθ; sin(θ+π)=−sinθ.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$(-11)\cos(4\pi/3)=11/2$
Apply the stated relation and retain its conditions.
$(-11)\sin(4\pi/3)=11\sqrt3/2$
The sign of the radius cancels the reversed direction.

The requested relation or conclusion is shown below.

x=11/2,\quad y=11\sqrt3/2

Checks and common pitfalls: The sign of the radius cancels the reversed direction.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

11 / Standard#Your turn

Describe the displayed polar locus with r≥0, and compare it with its full supporting line.

\theta=\pi/7,\quad r\ge0

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use both coordinate relations and retain the nonnegative radius.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$y=\tan(\pi/7)x, x\ge0$
The polar locus is a ray; dropping the sign restriction would add the opposite ray.

The requested relation or conclusion is shown below.

y=\tan(\pi/7)x,\quad x\ge0

Checks and common pitfalls: The polar locus is a ray; dropping the sign restriction would add the opposite ray.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

12 / Transfer#Your turn

Under the signed-radius convention, show that the two polar pairs represent the same point.

(13,\pi/3),\quad(-13,4\pi/3)

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use this intermediate relation.

cos(θ+π)=−cosθ; sin(θ+π)=−sinθ.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$(-13)\cos(4\pi/3)=13/2$
Apply the stated relation and retain its conditions.
$(-13)\sin(4\pi/3)=13\sqrt3/2$
The sign of the radius cancels the reversed direction.

The requested relation or conclusion is shown below.

x=13/2,\quad y=13\sqrt3/2

Checks and common pitfalls: The sign of the radius cancels the reversed direction.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

13 / Transfer#Your turn

Describe the displayed polar locus with r≥0, and compare it with its full supporting line.

\theta=\pi/2,\quad r\ge0

Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use both coordinate relations and retain the nonnegative radius.

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$x=0, y=r\ge0$
The polar locus is a ray; dropping the sign restriction would add the opposite ray.

The requested relation or conclusion is shown below.

x=0, y\ge0

Checks and common pitfalls: The polar locus is a ray; dropping the sign restriction would add the opposite ray.

Reasoning checklist · self / teacher assessment

State a valid definition or model and its assumptions.
Show the intermediate mathematical relations, not only the final claim.
Check exclusions, units or the interpretation of the result.

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

End-of-lesson check and correction

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Teacher preparation and assessment

Question sequence

Convert representations and eliminate polar variables without losing the geometric range.
Which condition is essential in polar coordinates?
Does adding π to an angle preserve the point when r stays positive?

Board plan

Model or definition: Convert representations and eliminate polar variables without losing the geometric range.
$x=r\cos\theta,\quad y=r\sin\theta$
Conditions: Angles are in radians; r≥0 is used unless a signed-radius convention is explicitly allowed. The origin has no unique polar angle.

Anticipated thinking

Equivalent angles differ by 2π; the same point may also have a signed negative-radius representation.

Assessment checklist

1 mark: choose the correct representation and conditions.
1 mark: establish the intermediate relation.
1 mark: complete a connected calculation or proof.
1 mark: interpret and check the conclusion.