LEARN · EXPLAIN · REVISE

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

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.

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

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

01 / Standard#Your turn

(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

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

Teacher preparation and assessment

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?

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.

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