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Polar Coordinates

Polar coordinates (r, θ) represent a point by distance r from the origin and angle θ from the positive x-axis. Conversion: r = √(x²+y²), θ = atan2(y,x); x = r cos θ, y = r sin θ. Used in physics, radar, and complex numbers.

Concept Fundamentals
r = √(x²+y²), θ = atan2(y,x)
To Polar
x = r cos θ, y = r sin θ
To Cartesian
r=1 → (cos θ, sin θ)
Unit Circle
atan2 handles all 4
Quadrants
Convert CoordinatesEnter Cartesian or polar values

Why This Mathematical Concept Matters

Why: Polar coordinates simplify circular motion, antenna patterns, and complex numbers (reiθ). atan2(y,x) correctly handles all quadrants, unlike atan(y/x).

How: From (x,y): r = √(x²+y²), θ = atan2(y,x). From (r,θ): x = r cos θ, y = r sin θ. Use radians for cos/sin; convert degrees with θ_rad = θ_deg × π/180.

  • ●atan2(y,x) gives correct angle in all quadrants.
  • ●r ≥ 0 convention; negative r = (|r|, θ+π).
  • ●Cylindrical coordinates = polar + z.
Sources:Khan Academy — Polar CoordinatesWolfram MathWorld — Polar Coordinates

Sample Examples

Input

Cartesian (x, y)

Polar (r, θ)

Results

Cartesian

(3, 4)

Polar

r = 5, θ = 53.13°

Visualization

⚠️For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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r = √(x²+y²), θ = atan2(y,x).

— Conversion

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Unit circle: (cos θ, sin θ).

— Trigonometry

Key Takeaways

  • r = √(x² + y²), θ = atan2(y, x)
  • x = r cos θ, y = r sin θ
  • θ is measured counterclockwise from positive x-axis
  • r ≥ 0; θ typically in [0, 2π)
  • Origin: r=0, θ undefined

Did You Know?

  • Polar coordinates simplify circular motion
  • atan2(y,x) handles all quadrants correctly
  • Used in radar, antenna patterns
  • Complex numbers: reiθ form
  • Polar plots common in engineering
  • Spherical coordinates extend to 3D

Understanding

Polar coordinates represent a point by distance r and angle θ.

r=x2+y2,θ=arctan⁡2(y,x)r = \sqrt{x^2 + y^2}, \quad \theta = \arctan2(y, x)
x=rcos⁡θ,y=rsin⁡θx = r \cos\theta, \quad y = r \sin\theta

Expert Tips

  • Use atan2, not atan(y/x), for correct quadrant
  • Normalize θ to [0, 2π) if needed
  • r=0: origin has no unique θ
  • Negative r can be represented as (r, θ+π)

FAQ

Q: Why atan2?
A: atan(y/x) fails in quadrants II and III.
Q: Can r be negative?
A: Conventionally r≥0; use θ for direction.
Q: Applications?
A: Physics, radar, complex numbers, graphics.
Q: Relation to cylindrical?
A: Cylindrical = polar + z.
Q: Degrees vs radians?
A: Radians standard in math; degrees for display.
Q: Multiple representations?
A: (r,θ) = (r,θ+2πk) for integer k.
Q: Origin?
A: r=0; θ arbitrary.

How to Use

  1. Enter Cartesian (x,y) or Polar (r,θ)
  2. Other form updates automatically
  3. θ in degrees or radians

Disclaimer

θ is in radians internally.

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