1. What Are Polar Coordinates?

Polar coordinates represent points in a plane using a distance from a reference point (r) and an angle from a reference direction (θ). Unlike Cartesian coordinates, each point can be represented by infinitely many polar coordinate pairs.

2. How to Find Equivalent Polar Coordinates

Any polar coordinate (r, θ) has equivalent representations:

Where:

• \( r \) — Magnitude (distance from origin)
• \( \theta \) — Angle in radians
• \( k \) — Any integer (positive, negative, or zero)

3. Understanding the Formula

Positive r form: Adding \( 2\pi k \) to θ gives equivalent coordinates because angles are periodic with period \( 2\pi \).
Negative r form: Flipping the sign of r and adding \( \pi \) to θ gives the same point in the opposite direction.

4. Using the Calculator

Tips: Enter the original r value (can be positive or negative), θ in radians, and any integer k. The calculator will show both equivalent forms.

5. Frequently Asked Questions (FAQ)

Q1: Why are there multiple representations?
A: Polar coordinates are periodic - adding full rotations (2π) doesn't change the point's location. Negative r with angle adjustment represents the same point.

Q2: What's the most common representation?
A: Typically r ≥ 0 and 0 ≤ θ < 2π, but other forms are mathematically equivalent.

Q3: How does k affect the result?
A: Each integer k gives another equivalent representation by adding full rotations.

Q4: Can θ be negative?
A: Yes, negative angles represent clockwise rotation from the reference direction.

Q5: What's the relationship to Cartesian coordinates?
A: Conversion formulas: \( x = r\cos\theta \), \( y = r\sin\theta \), and \( r = \sqrt{x^2 + y^2} \), \( \theta = \arctan(y/x) \).