Polar Coordinates
Instead of “go right, go up” (Cartesian), polar says “go this far, in this direction.” The coordinate uses distance and angle from the origin.
Introduction
Cartesian coordinates measure horizontal and vertical distances. Polar coordinates measure distance from origin and angle from positive x-axis. Some curves — like circles and spirals — are far simpler in polar form.
Past Knowledge
Cartesian coordinate plane, trig ratios, vector concept (7.8).
Today's Goal
Plot polar points, convert between polar and Cartesian, and recognize basic polar curves.
Future Success
Polar coordinates are essential for complex numbers, signal processing, and calculus in polar form.
Key Concepts
Conversion Formulas
| Polar → Cartesian | Cartesian → Polar |
|---|---|
Key Polar Curves
- → circle of radius
- → line through origin at angle
- or → circle passing through origin
Non-Unique Representation
Unlike Cartesian, polar coordinates are not unique. The point is the same as or . Adding or negating with a 180° shift gives the same point.
Worked Examples
Polar → Cartesian
Convert: to Cartesian.
Answer:
Cartesian → Polar
Convert: to polar.
Since the point is in Q2:
Answer:
Equation Conversion
Convert: to polar form.
Since :
Answer: (a circle of radius 3 centered at the origin)
Common Pitfalls
Wrong Quadrant for θ
returns values in Q1/Q4 only. For points in Q2 or Q3, add 180° to get the correct angle.
Assuming Unique Representation
A single point has infinitely many polar representations. When comparing points, convert to Cartesian or normalize the angle to .
Real-Life Applications
Radar Systems
Radar naturally operates in polar coordinates — it measures the distance (range) and angle (bearing) to detected objects. Air traffic control screens display aircraft positions as polar points, and converting to Cartesian is only done for map overlay purposes.
Practice Quiz
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