1. What Are Polar Coordinates?

Polar coordinates represent points in a plane using a distance from a reference point (radius r) and an angle from a reference direction (θ). This system is alternative to Cartesian coordinates and is particularly useful in circular and spiral patterns.

2. How Does the Calculator Work?

The calculator uses the polar to Cartesian conversion formula:

Where:

• \( r \) — Radial distance from origin (must be ≥0)
• \( \theta \) — Angle in degrees (measured counterclockwise from positive x-axis)
• \( x \) — Horizontal coordinate
• \( y \) — Vertical coordinate

Explanation: The formula converts polar coordinates to Cartesian coordinates using trigonometric functions.

3. Importance of Polar Coordinates

Details: Polar coordinates are essential in physics, engineering, and mathematics for problems involving circular symmetry, such as planetary motion, antenna radiation patterns, and complex number representation.

4. Using the Calculator

Tips: Enter the radius (r) as a non-negative number and angle (θ) in degrees. The calculator will provide the equivalent Cartesian (x,y) coordinates.

5. Frequently Asked Questions (FAQ)

Q1: Can r be negative in polar coordinates?
A: While mathematically possible, negative r values are typically avoided by adding 180° to θ instead.

Q2: How are angles measured?
A: Angles are measured in degrees, with 0° along the positive x-axis and increasing counterclockwise.

Q3: What's the range for θ?
A: θ can be any real number, but is often restricted to 0°-360° or -180° to 180° for simplicity.

Q4: When are polar coordinates better than Cartesian?
A: Polar coordinates are superior for problems with circular symmetry, like plotting orbits or designing circular structures.

Q5: How to convert back to polar from Cartesian?
A: Use \( r = \sqrt{x^2 + y^2} \) and \( \theta = \arctan(y/x) \) (with quadrant adjustment).