Polar Coordinates

Locating a point by a distance and an angle instead of two axes: the pole, the polar axis, how positive and negative angles turn, and why every polar point has infinitely many names.

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Polar Coordinates — Moosa Academy

Cartesian coordinates locate a point by how far across and how far up. Polar coordinates do the same job a different way: how far out, and at what angle.

Concept The pole and the polar axis
θ r P pole polar axis
The pole is the reference point — it does the same job as the origin. The polar axis is a single ray running to the right from the pole, replacing the two axes of the Cartesian system.
Cartesian: two axes, and a point is (x, y) .
Polar: one pole and one axis, and a point is (r, \theta) .
r is the distance from the pole; \theta is the angle from the polar axis.
Concept Which way the angle turns
A positive angle turns anticlockwise, upward from the polar axis.
A negative angle turns clockwise, downward from the polar axis.

Both start from the same place — the polar axis — and differ only in direction of rotation.

Example Plotting two points
(2, 30^\circ) — go 2 units out, turned 30° anticlockwise.
The point lands in the first quadrant.
(3, 200^\circ) — go 3 units out, turned 200° anticlockwise.
Past 180°, so the point lands in the third quadrant.

The distance is always written first and the angle second, so the order of the pair matters just as much as it does in Cartesian coordinates.

Concept One point, many names

A full turn is 360°, so adding or subtracting 360° lands on the very same spot. This is the one real difference from the Cartesian system, where each point has exactly one name.

(2, -30^\circ) turns 30° clockwise.
(2, 330^\circ) turns 330° anticlockwise.
Both arrive at the same point: -30^\circ + 360^\circ = 330^\circ .
Keep going and the names never run out:
(2, -30^\circ) , (2, 330^\circ) , (2, 690^\circ) , (2, 1050^\circ) …
⟹ every point has infinitely many polar representations
Summary
  1. Polar coordinates give a point as (r, θ): a distance and an angle.
  2. The pole replaces the origin; the polar axis is a single ray to the right.
  3. Positive angles turn anticlockwise, negative angles clockwise.
  4. Adding or subtracting 360° gives another name for the same point.
  5. So a polar point has infinitely many representations, unlike a Cartesian one.
231. Arithmetic Sequences and Series
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