Polar and Rectangular Form Calculator

Conversion Formulas:

\[ \text{Polar to Rectangular: } x = r \cdot \cos(\theta),\ y = r \cdot \sin(\theta) \] \[ \text{Rectangular to Polar: } r = \sqrt{x^2 + y^2},\ \theta = \arctan\left(\frac{y}{x}\right) \]

Magnitude (r):

unitless

Angle (θ):

radians

Real part (x):

unitless

Imaginary part (y):

unitless

Polar Form:

Rectangular Form:

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1. What Are Polar and Rectangular Forms?

The polar form (r ∠ θ) and rectangular form (x + yi) are two ways to represent complex numbers. Polar form uses magnitude and angle, while rectangular form uses real and imaginary components.

2. How Does the Calculator Work?

The calculator uses these conversion formulas:

\[ \text{Polar to Rectangular: } x = r \cdot \cos(\theta),\ y = r \cdot \sin(\theta) \] \[ \text{Rectangular to Polar: } r = \sqrt{x^2 + y^2},\ \theta = \arctan\left(\frac{y}{x}\right) \]

Where:

\( r \) — Magnitude (always ≥ 0)
\( \theta \) — Angle in radians
\( x \) — Real component
\( y \) — Imaginary component
\( i \) — Imaginary unit (√-1)

Explanation: The calculator automatically detects which conversion to perform based on which fields you fill in.

3. Importance of Complex Number Forms

Details: Polar form is better for multiplication/division, while rectangular form is better for addition/subtraction. Different engineering applications prefer different forms.

4. Using the Calculator

Tips: Enter either polar coordinates (r and θ) or rectangular coordinates (x and y). The calculator will compute the other form. Angle θ should be in radians.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between degrees and radians?
A: Radians are the standard mathematical unit (2π = 360°). For degrees, convert first (θ_rad = θ_deg × π/180).

Q2: How do I represent negative r values?
A: Magnitude r should always be ≥ 0. For negative r, add π to θ instead.

Q3: What's the principal value of θ?
A: The calculator returns θ in the range (-π, π] using atan2 function.

Q4: Can I use this for 3D coordinates?
A: No, this is for 2D complex numbers only. For 3D, you need spherical coordinates.

Q5: Why might results differ slightly from other calculators?
A: Small differences may occur due to floating-point precision and different rounding methods.