Lesson 1: The Polar Coordinate System
Estimated time: 30-35 minutes
Learning Objectives
- Define polar coordinates (r, θ) and contrast with rectangular (x, y)
- Plot points in polar coordinates
- Find multiple representations of the same point
- Understand the role of negative r values
What Are Polar Coordinates?
Polar Coordinates (r, θ) locate a point by its distance r from the origin (pole) and the angle θ from the positive x-axis (polar axis).
Unlike rectangular coordinates where each point has a unique (x, y), in polar a single point has infinitely many representations: (r, θ), (r, θ+2nπ), and (−r, θ+π) all represent the same point.
Plotting Points
Example 1: Plot (3, π/4)
Move 3 units from the origin in the direction π/4 (45°). This is in QI.
Example 2: Plot (2, 5π/6)
Move 2 units from the origin at angle 5π/6 (150°). This is in QII.
Example 3: Negative r
Plot (−2, π/3). A negative r means go in the opposite direction of π/3. Move 2 units in the direction π/3 + π = 4π/3. This lands in QIII.
Multiple Representations
Example 4: Find Three Representations of (4, π/6)
1. (4, π/6) — original
2. (4, π/6 + 2π) = (4, 13π/6) — add full rotation
3. (−4, π/6 + π) = (−4, 7π/6) — negate r, add π
Special Points
The pole (origin) is represented as (0, θ) for any angle θ.
Points on the polar axis (θ = 0) have form (r, 0).
Check Your Understanding
1. Plot (5, 3π/2). Which axis does this point lie on?
2. Give another representation of (3, π/4) with negative r.
3. Where is (−2, 0) in polar coordinates?
Key Takeaways
- Polar coordinates (r, θ) specify distance from origin and angle from polar axis.
- Each point has infinitely many polar representations.
- Negative r means move in the opposite direction of angle θ.
- The pole (origin) is (0, θ) for any θ.