Polar Coordinates

Polar Coordinate System

Alternative to rectangular (x, y) coordinates

Polar coordinates: (r, θ)

• r = distance from origin (radius)
• θ = angle from positive x-axis (counterclockwise)

Origin called: Pole

Positive x-axis called: Polar axis

Example: Plot Points

Point (3, 60°):

• Move 3 units from origin
• At angle 60° from positive x-axis

Point (2, π):

• Move 2 units from origin
• At angle π (180°) = negative x-axis
• Location: (-2, 0) in rectangular

Converting Polar to Rectangular

Formulas:

x = r·cos(θ)

y = r·sin(θ)

Derived from right triangle in unit circle

Example 1: Convert to Rectangular

Polar: (4, 60°)

Calculate:

x = 4·cos(60°) = 4(1/2) = 2
y = 4·sin(60°) = 4(√3/2) = 2√3

Rectangular: (2, 2√3)

Example 2: With Radians

Polar: (6, π/3)

Calculate:

x = 6·cos(π/3) = 6(1/2) = 3
y = 6·sin(π/3) = 6(√3/2) = 3√3

Rectangular: (3, 3√3)

Example 3: Negative Radius

Polar: (-2, 0)

Negative r means opposite direction:

x = -2·cos(0) = -2(1) = -2
y = -2·sin(0) = -2(0) = 0

Rectangular: (-2, 0)

Converting Rectangular to Polar

Formulas:

r = √(x² + y²) or r = ±√(x² + y²)

tan(θ) = y/x, so θ = tan⁻¹(y/x)

Must check quadrant for correct θ!

Example 1: First Quadrant

Rectangular: (3, 4)

Find r:

r = √(3² + 4²) = √25 = 5

Find θ:

tan(θ) = 4/3
θ = tan⁻¹(4/3) ≈ 53.1°

Polar: (5, 53.1°) or (5, 0.927 rad)

Example 2: Second Quadrant

Rectangular: (-2, 2)

Find r:

r = √(4 + 4) = √8 = 2√2

Find θ:

tan(θ) = 2/(-2) = -1
θ = tan⁻¹(-1) = -45°

But point in Q2, so:

θ = 180° - 45° = 135°

Or: θ = 3π/4

Polar: (2√2, 135°)

Example 3: Third Quadrant

Rectangular: (-3, -3)

Find r:

r = √(9 + 9) = 3√2

Find θ:

tan⁻¹(-3/-3) = tan⁻¹(1) = 45°

But in Q3:

θ = 180° + 45° = 225°

Polar: (3√2, 225°) or 3√2, 5π/4)

Multiple Representations

Same point has infinitely many polar representations

(r, θ) = (r, θ + 2πn) for any integer n

Also: (r, θ) = (-r, θ + π)

Example: Equivalent Forms

Point (3, π/4):

Also written as:

• (3, π/4 + 2π) = (3, 9π/4)
• (3, π/4 - 2π) = (3, -7π/4)
• (-3, π/4 + π) = (-3, 5π/4)

All represent same point!

Polar Equations

Equation relating r and θ

Examples:

• r = 2 (circle)
• θ = π/3 (line)
• r = 2sin(θ) (circle)

Graphing Polar Equations

Create table of θ and r values

Plot points (r, θ)

Connect to form curve

Example 1: Circle

r = 3

All points 3 units from origin

Circle of radius 3 centered at origin

Example 2: Line Through Origin

θ = π/4

All points at angle 45°

Line through origin at 45° angle

Example 3: Spiral

r = θ for θ ≥ 0

Distance increases with angle → spiral

Common Polar Curves

Circle through origin:

• r = 2a·cos(θ) (horizontal)
• r = 2a·sin(θ) (vertical)

Cardioid: r = a(1 + cos(θ))

Rose curve: r = a·cos(nθ)

• n petals if n odd
• 2n petals if n even

Lemniscate: r² = a²·cos(2θ)

Example: Cardioid

r = 2(1 + cos(θ))

Heart-shaped curve

Example: Rose Curve

r = 3·sin(2θ)

4 petals (n = 2, so 2n = 4 petals)

Maximum r = 3 when sin(2θ) = 1

Converting Equations

Polar to rectangular: Substitute r² = x² + y², x = r cos θ, y = r sin θ

Rectangular to polar: Substitute x = r cos θ, y = r sin θ

Example 1: Polar to Rectangular

r = 4sin(θ)

Multiply by r:

r² = 4r·sin(θ)

Substitute:

x² + y² = 4y

Complete square:

x² + y² - 4y = 0
x² + (y - 2)² = 4

Circle centered at (0, 2) with radius 2

Example 2: Rectangular to Polar

x² + y² = 16

Recognize: r² = x² + y²

Polar form: r² = 16

Or: r = ±4

Circle of radius 4

Example 3: Line

Rectangular: y = x

Since y/x = tan(θ):

tan(θ) = 1
θ = π/4

Polar: θ = π/4 (line through origin at 45°)

Symmetry in Polar Coordinates

Symmetry about x-axis: Replace θ with -θ

Symmetry about y-axis: Replace θ with π - θ

Symmetry about origin: Replace r with -r

Example: Test Symmetry

r = 4cos(θ)

Test x-axis symmetry: Replace θ with -θ

r = 4cos(-θ) = 4cos(θ)

Same equation → symmetric about x-axis!

Real-World Applications

Navigation: Bearings and distances

Radar: Distance and angle to target

Astronomy: Celestial coordinates

Engineering: Antenna patterns, stress distribution

Biology: Spiral shells, flower petals

Example: Navigation

Ship is 5 nautical miles at bearing 120° from harbor

Polar: (5, 120°)

Convert to rectangular (relative to harbor):

x = 5·cos(120°) = 5(-1/2) = -2.5
y = 5·sin(120°) = 5(√3/2) ≈ 4.33

2.5 nm west, 4.33 nm north of harbor

Advantages of Polar Coordinates

Natural for circular/spiral patterns

Simpler equations for certain curves

Useful when problem has radial symmetry

Practice