Polar Coordinates — Complete Guide
The polar coordinate system, converting between polar and rectangular, symmetry tests, and graphing all major polar curve types with worked examples.
Quick Reference — Conversion Formulas
The Polar Coordinate System
In polar coordinates, a point is described by (r, θ) where r is the distance from the origin (pole) and θ is the angle from the positive x-axis (polar axis).
Positive r
Start at the pole. Rotate to angle θ (counterclockwise for positive θ). Move r units in that direction.
(3, π/4) → rotate 45°, go 3 units
Negative r
Rotate to angle θ, then move |r| units in the opposite direction (equivalent to adding π to the angle).
(−3, π/4) = (3, 5π/4)
Multiple Representations — Key Fact
Unlike rectangular coordinates, every polar point has infinitely many representations:
Converting Coordinates — Worked Examples
Rectangular → Polar
Convert (−3, 3) to polar (r > 0, 0 ≤ θ < 2π)
r = √(9 + 9) = √18 = 3√2
reference angle: arctan(3/3) = arctan(1) = π/4
Quadrant II (x < 0, y > 0): θ = π − π/4 = 3π/4
Answer: (3√2, 3π/4)
Polar → Rectangular
Convert (4, 2π/3) to rectangular
x = 4 cos(2π/3) = 4(−1/2) = −2
y = 4 sin(2π/3) = 4(√3/2) = 2√3
Answer: (−2, 2√3)
Convert an Equation: r = 3 / (1 − 2 sin θ) to Rectangular
Step 1: Multiply both sides by (1 − 2 sin θ): r − 2r sin θ = 3
Step 2: Substitute r sin θ = y: r − 2y = 3 → r = 3 + 2y
Step 3: Square both sides: r² = (3 + 2y)²
Step 4: Replace r² = x² + y²: x² + y² = (3 + 2y)² = 9 + 12y + 4y²
x² − 3y² + 12y + 9 = 0 (hyperbola)
Symmetry Tests for Polar Curves
| Axis of Symmetry | Test |
|---|---|
| Polar axis (x-axis) | Replace θ with −θ. If the equation is unchanged, symmetric about the polar axis. |
| Line θ = π/2 (y-axis) | Replace θ with π − θ. If the equation is unchanged, symmetric about θ = π/2. |
| Pole (origin) | Replace r with −r (or θ with θ + π). If unchanged, symmetric about the pole. |
Note: Passing a symmetry test guarantees symmetry; failing does not guarantee asymmetry (because multiple representations can still produce symmetric graphs).
Major Polar Curve Types
Circle (centered at origin)
All points at constant distance a from the pole.
r = 3 → circle of radius 3
Circle (through origin)
Diameter 2a, passes through the pole.
r = 4 cos θ → circle, center (2, 0), radius 2
Cardioid
Heart-shaped. Special case of limaçon where a = b.
r = 2(1 + cos θ)
Limaçon
Cardioid when a=b; inner loop when a<b; convex when a≥2b.
r = 2 + 3 cos θ (inner loop, a < b)
Rose Curve
n petals if n is odd; 2n petals if n is even.
r = 4 cos(3θ) → 3 petals; r = 3 cos(2θ) → 4 petals
Lemniscate
Figure-eight curve (∞ shape). Only exists where r² ≥ 0.
r² = 9 cos(2θ) → figure-eight, a = 3
Rose Curves — Petal Count Rule
For r = a cos(nθ) or r = a sin(nθ):
Common mistake: Students count petals incorrectly for even n. r = 4 cos(2θ) has 4 petals (2×2), not 2. Always double n when n is even.
Frequently Asked Questions
How do you convert between polar and rectangular coordinates?
Use these four relationships: x = r·cos θ, y = r·sin θ, r² = x² + y², and tan θ = y/x. To convert polar (r, θ) to rectangular (x, y): plug into x = r cos θ and y = r sin θ. To convert rectangular (x, y) to polar: find r = √(x² + y²) and θ = arctan(y/x) adjusted for the correct quadrant (check the signs of x and y).
What are the main types of polar curves?
The main polar curve types are: circles (r = a, or r = a·cos θ / r = a·sin θ), limaçons (r = a ± b·cos θ or a ± b·sin θ — cardioids when a=b, inner loop when a<b, convex when a≥2b), rose curves (r = a·cos(nθ) or a·sin(nθ) — n petals if n is odd, 2n petals if n is even), and lemniscates (r² = a²·cos 2θ).
Can a point have more than one polar representation?
Yes — every point in the polar plane has infinitely many representations. The point (r, θ) is equivalent to (r, θ + 2πk) for any integer k (full rotations), and also to (−r, θ + π) (going the opposite direction). For example, (2, π/3) = (2, π/3 + 2π) = (−2, 4π/3). This non-uniqueness is important when checking if two polar equations pass through the same point.
Practice polar coordinate problems
Converting coordinates, identifying curve types, graphing polar equations — all with step-by-step solutions. Free to start.
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