Rectangular Form to Polar Form Calculator

Conversion Formulas:

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

Magnitude (r):

Angle (θ):

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1. What is Rectangular to Polar Conversion?

The rectangular to polar conversion transforms coordinates from Cartesian (x,y) form to polar (r,θ) form. This is particularly useful in fields like physics, engineering, and mathematics where polar coordinates can simplify problems involving circular or rotational symmetry.

2. How Does the Calculator Work?

The calculator uses the following formulas:

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

Where:

\( r \) — Magnitude (distance from origin)
\( \theta \) — Angle (in radians) from positive x-axis
\( x \) — X coordinate in Cartesian system
\( y \) — Y coordinate in Cartesian system

Explanation: The magnitude (r) is calculated using the Pythagorean theorem, while the angle (θ) is calculated using the arctangent function (atan2 in implementation for correct quadrant determination).

3. Importance of Polar Coordinates

Details: Polar coordinates are essential for solving problems involving circular motion, wave propagation, and complex numbers. They often simplify equations in these domains compared to rectangular coordinates.

4. Using the Calculator

Tips: Enter both x and y coordinates. The calculator will automatically compute the corresponding polar coordinates. Both inputs are unitless, and the results will be in unitless magnitude and radians for the angle.

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, unlike atan(y/x) which only returns values between -π/2 and π/2.

Q2: How do I convert the angle to degrees?
A: Multiply the angle in radians by 180/π (approximately 57.2958). Many calculators have a radians-to-degrees conversion function.

Q3: What happens when x and y are both zero?
A: The origin (0,0) in rectangular coordinates corresponds to r=0 in polar coordinates, but the angle θ is undefined in this case.

Q4: Can negative r values be used in polar coordinates?
A: While r is typically non-negative, some conventions allow negative r values which are interpreted as being in the opposite direction of the angle θ.

Q5: Where are polar coordinates commonly used?
A: Polar coordinates are widely used in physics (especially electromagnetism), engineering (antenna radiation patterns), navigation, and complex number analysis.