1. What is Polar to Cartesian Conversion?

The polar to Cartesian coordinate conversion transforms coordinates from the polar system (r, θ) to the Cartesian system (x, y). This is essential in mathematics, physics, engineering, and computer graphics where different coordinate systems are used.

2. How Does the Calculator Work?

The calculator uses the standard conversion formulas:

Where:

• \( r \) — Radius (distance from origin)
• \( \theta \) — Angle in radians (measured from positive x-axis)
• \( x \) — Horizontal coordinate
• \( y \) — Vertical coordinate

Explanation: The formulas project the polar coordinates onto the x and y axes using trigonometric functions.

3. Importance of Coordinate Conversion

Details: Many physical systems are naturally described in polar coordinates (like circular motion), while most calculations and visualizations are done in Cartesian coordinates. Conversion between these systems is fundamental in many fields.

4. Using the Calculator

Tips: Enter the radius (must be ≥0) and angle in radians. For degrees, convert to radians first (multiply by π/180). The calculator will output the corresponding x and y coordinates.

5. Frequently Asked Questions (FAQ)

Q1: Why use radians instead of degrees?
A: Radians are the natural unit for angle measurement in mathematics, making derivatives and other operations simpler. Most programming languages and mathematical functions use radians.

Q2: What's the range for valid θ values?
A: The calculator accepts any real number for θ, as trigonometric functions are periodic with period 2π. The result will be equivalent to θ modulo 2π.

Q3: How do negative radii work?
A: Negative radii are equivalent to adding π to the angle and using the absolute value of the radius. However, this calculator only accepts non-negative radii.

Q4: What's the relationship to complex numbers?
A: Polar coordinates correspond to the modulus and argument of a complex number, while Cartesian coordinates correspond to its real and imaginary parts.

Q5: How is this used in Desmos?
A: Desmos can graph in both coordinate systems. Understanding conversion helps when switching between polar and Cartesian graphing modes.