How To Convert Rectangular To Polar In Calculator

Rectangular to Polar Conversion Formulas:

\[ r = \sqrt{x^2 + y^2} \] \[ \theta = \arctan\left(\frac{y}{x}\right) \]

Radius (r):

Angle (θ):

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1. What Is Rectangular To Polar Conversion?

Rectangular to polar coordinate conversion transforms Cartesian coordinates (x, y) into polar coordinates (r, θ). This is essential in mathematics, physics, and engineering for problems involving circular symmetry or angular relationships.

2. How Does The Calculator Work?

The calculator uses the standard conversion formulas:

\[ r = \sqrt{x^2 + y^2} \] \[ \theta = \arctan\left(\frac{y}{x}\right) \]

Where:

\( r \) — Radius or distance from origin
\( \theta \) — Angle from positive x-axis
\( x \) — Horizontal coordinate
\( y \) — Vertical coordinate

Explanation: The radius is calculated using the Pythagorean theorem, while the angle is determined using the arctangent function, which properly handles all quadrants.

3. Importance Of Coordinate Conversion

Details: Polar coordinates are particularly useful in fields like electromagnetism, signal processing, and navigation where circular patterns or rotational symmetry are involved. They simplify many mathematical operations and physical interpretations.

4. Using The Calculator

Tips: Enter x and y coordinates in any consistent units. Select whether you want the angle result in radians (mathematical standard) or degrees (common in engineering). The calculator handles all quadrants automatically.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of the angle θ?
A: The angle typically ranges from -π to π radians (-180° to 180°) or 0 to 2π radians (0° to 360°), depending on the convention used.

Q2: How does the calculator handle negative coordinates?
A: The calculator uses atan2 function which properly determines the quadrant based on the signs of both x and y coordinates.

Q3: What if both x and y are zero?
A: At the origin (0,0), the radius is 0 but the angle is undefined. The calculator will show appropriate results.

Q4: When should I use radians vs degrees?
A: Use radians for mathematical calculations and calculus, degrees for engineering applications and everyday measurements.

Q5: Can I convert back from polar to rectangular?
A: Yes, using the formulas: x = r·cos(θ) and y = r·sin(θ).