Adding Polar Form Calculator

Polar Addition Formula:

\[ z_1 + z_2 = r_1 (\cos \theta_1 + i \sin \theta_1) + r_2 (\cos \theta_2 + i \sin \theta_2) \]

Magnitude (r₁):

unitless

Angle (θ₁):

radians

Magnitude (r₂):

unitless

Angle (θ₂):

radians

Rectangular Form Result:

Polar Form Result:

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1. What is Polar Form Addition?

Adding complex numbers in polar form involves converting them to rectangular form first, performing the addition, and then converting the result back to polar form. This method is necessary because addition in polar coordinates isn't straightforward.

2. How Does the Calculator Work?

The calculator uses the following steps:

\[ \text{1. Convert to rectangular: } z_1 = r_1\cos\theta_1 + ir_1\sin\theta_1 \] \[ z_2 = r_2\cos\theta_2 + ir_2\sin\theta_2 \] \[ \text{2. Add components: } z_1 + z_2 = (x_1+x_2) + i(y_1+y_2) \] \[ \text{3. Convert back to polar: } r = \sqrt{x^2 + y^2}, \theta = \tan^{-1}(y/x) \]

3. Importance of Polar Form Addition

Details: While polar form is excellent for multiplication and division, addition requires rectangular form. This conversion is essential in electrical engineering, signal processing, and physics.

4. Using the Calculator

Tips: Enter magnitudes (always positive) and angles in radians. The calculator will show both rectangular and polar form results.

5. Frequently Asked Questions (FAQ)

Q1: Why can't we add polar forms directly?
A: Polar coordinates represent magnitude and angle, which don't add linearly. Rectangular coordinates (real and imaginary parts) add component-wise.

Q2: What's the advantage of polar form?
A: Polar form simplifies multiplication (multiply magnitudes, add angles) and division (divide magnitudes, subtract angles).

Q3: How to convert degrees to radians?
A: Multiply degrees by π/180. Many calculators have a degree-to-radian conversion function.

Q4: What's the range for the angle θ?
A: The calculator uses atan2 which gives angles in the range (-π, π] radians.

Q5: Can I add more than two complex numbers?
A: Yes, convert all to rectangular form, add all components, then convert the sum back to polar.