1. What is Polar Coordinate Conversion?

Polar coordinate conversion transforms Cartesian (x,y) coordinates into polar (r,θ) coordinates, where r is the distance from the origin and θ is the angle from the positive x-axis. This is useful in many physics and engineering applications.

2. How Does the Calculator Work?

The calculator uses the following equations:

Where:

• \( x \) — Horizontal coordinate
• \( y \) — Vertical coordinate
• \( r \) — Distance from origin (always positive)
• \( \theta \) — Angle in radians (from -π to π)

Explanation: The calculator uses atan2(y,x) which properly determines the quadrant of the angle, unlike a simple arctan(y/x) calculation.

3. Importance of Polar Coordinates

Details: Polar coordinates are essential in many fields including physics (especially for circular motion), engineering (antenna radiation patterns), and mathematics (complex numbers). They often simplify problems with circular symmetry.

4. Using the Calculator

Tips: Enter Cartesian x and y coordinates (unitless values). The calculator will output the corresponding polar coordinates r (radius) and θ (angle in radians). For (0,0), the angle is undefined.

5. Frequently Asked Questions (FAQ)

Q1: Why use polar coordinates instead of Cartesian?
A: Polar coordinates are more natural for problems involving circular symmetry, rotations, or periodic phenomena.

Q2: How do I convert the angle to degrees?
A: Multiply the radians value by (180/π). Many calculators have a radians-to-degrees conversion function.

Q3: What's the range of the angle θ?
A: The angle ranges from -π to π radians (-180° to 180°), with positive angles measured counterclockwise from the positive x-axis.

Q4: What happens at the origin (0,0)?
A: The radius r is 0, but the angle θ is undefined since there's no meaningful direction at the origin.

Q5: Can I convert back to Cartesian coordinates?
A: Yes! Use x = r·cos(θ) and y = r·sin(θ) to convert from polar back to Cartesian coordinates.