1. What is Cartesian to Polar Conversion?

Cartesian to polar conversion is a mathematical transformation that converts coordinates from the rectangular (x,y) system to the polar (r,θ) system. This is particularly useful in fields like physics, engineering, and navigation where polar coordinates can simplify problems involving circular or rotational symmetry.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Where:

• \( x \) — Cartesian x-coordinate
• \( y \) — Cartesian y-coordinate
• \( r \) — Polar radius (distance from origin)
• \( \theta \) — Polar angle in radians (measured counter-clockwise from positive x-axis)

Explanation: The radius (r) is calculated using the Pythagorean theorem, while the angle (θ) is calculated using the atan2 function which properly handles all four quadrants.

3. Importance of Coordinate Conversion

Details: Polar coordinates are essential for solving problems involving circular motion, wave propagation, and any system with radial symmetry. Many physical systems are more naturally described in polar coordinates than Cartesian coordinates.

4. Using the Calculator

Tips: Simply enter the x and y coordinates in the input fields. The calculator will compute the corresponding polar coordinates (r, θ). The angle is returned in radians between -π and π.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between atan and atan2?
A: atan2(y,x) is preferred as it considers the signs of both coordinates to determine the correct quadrant of the angle, while atan(y/x) only gives results between -π/2 and π/2.

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 if my point is at the origin (0,0)?
A: The radius will be 0, but the angle is technically undefined at the origin.

Q4: Are there limitations to this conversion?
A: While mathematically precise, be aware of floating-point precision limitations in computer calculations, especially for very large or very small numbers.

Q5: Can I convert back to Cartesian coordinates?
A: Yes! The reverse conversion is x = r·cos(θ) and y = r·sin(θ).