What is a polar coordinate pair?

An ordered pair $(r, φ)$ giving a point's radial distance from the pole and angular direction from the polar axis.

What are the pole and polar axis?

The pole is the reference point from which distance and direction are measured. The polar axis is the reference ray from the pole used to measure angles.

How are degrees and radians related?

A full revolution is $360° = 2π$ radians, and a half revolution is $180° = π$ radians.

How can an angle be changed without changing its direction?

Add or subtract any integer number of full turns: $φ$ and $φ + 360°n$ (or $φ + 2πn$) point in the same direction.

Why is a polar representation not always unique?

Full turns leave direction unchanged, negative radii can be replaced by a half-turn angle change, and every angle with $r=0$ denotes the pole.

CalculusLesson 2 of 5 in the course Parametric and polar curves

Polar Coordinates: Geometry, Complex Numbers, Calculus, and Applications

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

Polar coordinates describe a point using its distance and direction from a fixed center. This lesson introduces the pole, radial coordinate, polar angle, and the conventions used to interpret angles.

The polar coordinate system locates a point in a plane by measuring from a reference point called the pole. The pole plays a role similar to the origin in Cartesian coordinates. A ray extending from the pole establishes the polar axis.

All flashcards

What is a polar coordinate pair?: An ordered pair $(r, φ)$ giving a point's radial distance from the pole and angular direction from the polar axis.
What are the pole and polar axis?: The pole is the reference point from which distance and direction are measured. The polar axis is the reference ray from the pole used to measure angles.
How are degrees and radians related?: A full revolution is $360° = 2π$ radians, and a half revolution is $180° = π$ radians.
How can an angle be changed without changing its direction?: Add or subtract any integer number of full turns: $φ$ and $φ + 360°n$ (or $φ + 2πn$ ) point in the same direction.
Why is a polar representation not always unique?: Full turns leave direction unchanged, negative radii can be replaced by a half-turn angle change, and every angle with $r=0$ denotes the pole.

Quiz questions

What does the radial coordinate measure?Answer: The signed distance from the pole to the point.
What is the pole analogous to in Cartesian coordinates?Answer: The origin.
In the usual mathematical convention, which way does a positive angle rotate?Answer: Counterclockwise from the rightward polar axis.
Which pair represents the same point as $(3, 40°)$ using a negative radius?Answer: $(-3, 220°)$
Why can $(0, φ)$ represent the pole for any angle $φ$ ?Answer: A zero radial distance places the point at the pole regardless of direction.

Based on the Wikipedia article Polar coordinate system, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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