Convert a complex number from polar form (r, θ) to rectangular form a + bi. Enter the modulus r and argument θ in either degrees or radians. The calculator applies the formulas a = r cos(θ) and b = r sin(θ) to give you the real and imaginary parts instantly.
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Standard Formula
Rectangular form conversion: a = r cos(θ), b = r sin(θ). Valid for any complex number in polar form.
Polar Form Input
Please enter a valid non-negative number for r.
Please enter a valid number for θ.
Rectangular Form Result
The rectangular form a + bi where:
Real part (a = r cos θ)
3.000
Imaginary part (b = r sin θ)
4.000
Conversion Result
Real Part (a)
3.000
r cos(θ)
Rectangular Form
3 + 4i
a + bi
Imaginary Part (b)
4.000
r sin(θ)
Step-by-Step Workings
Modulus (r)5
Argument (θ)53.13°
Argument in radians0.9274 rad
cos(θ)0.6000
sin(θ)0.8000
a = r × cos(θ)5 × 0.6000 = 3.000
b = r × sin(θ)5 × 0.8000 = 4.000
Rectangular form3.000 + 4.000i
Worked Example (Default Values)
Convert the complex number with modulus r = 5 and argument θ = 53.13° to rectangular form.
Step 1 - Find the real part: a = r cos(θ) = 5 × cos(53.13°) = 5 × 0.6000 = 3.000
Step 2 - Find the imaginary part: b = r sin(θ) = 5 × sin(53.13°) = 5 × 0.8000 = 4.000
Result: The rectangular form is 3.000 + 4.000i
This is the classic 3-4-5 right triangle example: the complex number 3 + 4i has modulus 5 (since √(3² + 4²) = 5) and argument arctan(4/3) = 53.13°.
How to Convert Polar Form to Rectangular Form
A complex number can be written in two equivalent ways. In polar form it is expressed as r(cos θ + i sin θ), or more compactly as r∠θ, where r is the modulus (the distance from the origin on the complex plane) and θ is the argument (the angle measured anticlockwise from the positive real axis). In rectangular form the same number is written as a + bi, where a is the real part and b is the imaginary part.
The conversion formulas come directly from trigonometry. If you draw the complex number on the Argand diagram, it forms a right triangle with the real axis, so:
Component
Formula
Description
Real part (a)
a = r cos(θ)
Horizontal component
Imaginary part (b)
b = r sin(θ)
Vertical component
The angle θ can be given in degrees or radians. To convert from degrees to radians, multiply by π/180. This calculator accepts both.
Common Conversion Examples
Polar Form (r, θ in degrees)
Real Part (a)
Imaginary Part (b)
Rectangular Form
r = 1, θ = 0°
1.000
0.000
1 + 0i
r = 1, θ = 90°
0.000
1.000
0 + 1i
r = 1, θ = 180°
-1.000
0.000
-1 + 0i
r = 1, θ = 270°
0.000
-1.000
0 - 1i
r = 5, θ = 53.13°
3.000
4.000
3 + 4i
r = 2, θ = 45°
1.414
1.414
1.414 + 1.414i
r = 3, θ = 30°
2.598
1.500
2.598 + 1.500i
When to Use Each Form
Rectangular form (a + bi) is most convenient for addition and subtraction of complex numbers: you simply add or subtract the real parts and the imaginary parts separately. For example, (3 + 4i) + (1 + 2i) = 4 + 6i.
Polar form (r, θ) is most convenient for multiplication and division. To multiply two complex numbers in polar form, multiply their moduli and add their arguments: r1 * r2 at angle θ1 + θ2. To divide, divide the moduli and subtract the arguments.
Powers and roots are also straightforward in polar form using De Moivre's theorem: (r∠θ)^n = r^n ∠ nθ.
The Euler and Cis Notation
A complex number in polar form can also be written using Euler's formula: r e^(iθ) = r(cos θ + i sin θ). The shorthand cis(θ) means cos(θ) + i sin(θ), so r∠θ = r cis(θ) = r e^(iθ). All three notations describe the same complex number, and the rectangular form a + bi is obtained from any of them using a = r cos θ and b = r sin θ.
Precision Note
Due to floating-point arithmetic, values that are mathematically zero (such as the imaginary part at θ = 0° or 180°) may appear as very small numbers like 0.000 or -0.000. This calculator rounds results to three decimal places to keep the output clean. For exact symbolic results, treat any value below 1×10⁻¹⁰ as zero.
Sources and method: Conversion formulas from standard complex analysis: a = r cos(θ), b = r sin(θ), derived from Euler's formula and the definition of polar coordinates on the complex plane. See Kreyszig, Advanced Engineering Mathematics, 10th ed., Chapter 13.
This calculator computes results to floating-point precision and rounds to three decimal places. For angles that are exact multiples of 30° or 45°, the results are exact to many more decimal places than displayed. Always verify critical computations independently.
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