BIPM-ratified constants · v1.0

Converter

Complex, number polar to rectangular form converter calculator.

Convert polar form complex numbers (r∠θ) to rectangular form (a + bi) using a = r·cosθ and b = r·sinθ. Supports degrees and radians.

From

degrees (°)

degrees

Angle (θ)

5 degrees =3Rectangular Component (a or b in a+bi)

Equivalents

Precision: 6 dp · Notation: Decimal · 3 units

Real part (a)real3

Imaginary part (b)imaginary4

Verify magnitude √(a²+b²)verify5

Common pairings

1 degreesequals0.6 real

1 degreesequals0.8 imaginary

1 degreesequals1 verify

1 radiansequals-0.961887 real

1 radiansequals0.273447 imaginary

1 radiansequals1 verify

The conversion

How the value
is computed.

What Is Polar-to-Rectangular Form Conversion?

A complex number in polar form is defined by two quantities: its magnitude r (the modulus, or distance from the origin in the complex plane) and its angle θ (the argument, measured counter-clockwise from the positive real axis). Converting to rectangular form expresses that same number as a + bi, where a is the real part and b is the imaginary part. This complex number to rectangular form converter applies standard trigonometric identities to perform the conversion accurately for both degree and radian inputs.

The Conversion Formula

The fundamental relationship connecting polar and rectangular forms is:

z = r(cos θ + i sin θ) = a + bi

The two rectangular components are extracted as:

a = r cos θ — Real part: the projection of the magnitude onto the horizontal (real) axis
b = r sin θ — Imaginary part: the projection of the magnitude onto the vertical (imaginary) axis

This formula derives directly from right-triangle trigonometry applied in the complex plane. A point at distance r from the origin at angle θ has Cartesian coordinates (r cos θ, r sin θ), which map directly to the real and imaginary components of the complex number.

Degrees vs. Radians

The calculator accepts angles in either degrees or radians. When degrees are selected, the conversion θ_rad = θ_deg × (π / 180) is applied before computing the trigonometric functions. Standard mathematical environments operate on radians internally — for example, 60° becomes π/3 ≈ 1.0472 rad. Selecting the wrong unit produces incorrect results: entering 90 with radians selected yields cos(90 rad) ≈ −0.448 rather than 0. Always confirm the unit before computing.

Magnitude Check

Beyond displaying the real and imaginary parts individually, the calculator offers a magnitude check output. After converting to rectangular form, it computes √(a² + b²) and compares the result against the original r. A correct conversion recovers the original modulus exactly. Discrepancies on the order of 10⁻¹⁵ are normal IEEE 754 double-precision floating-point rounding effects and do not indicate an error in the conversion.

Worked Examples

Example 1: 45° (First Quadrant)

Convert z = 5∠45° to rectangular form.

a = 5 × cos(45°) = 5 × 0.70711 ≈ 3.5355
b = 5 × sin(45°) = 5 × 0.70711 ≈ 3.5355
Result: z ≈ 3.5355 + 3.5355i

Example 2: 120° (Second Quadrant, Negative Real Part)

Convert z = 10∠120° to rectangular form.

a = 10 × cos(120°) = 10 × (−0.5) = −5.000
b = 10 × sin(120°) = 10 × 0.86603 ≈ 8.660
Result: z = −5 + 8.660i

Example 3: Radian Input (π/6)

Convert z = 4∠(π/6) to rectangular form.

a = 4 × cos(π/6) = 4 × 0.86603 ≈ 3.464
b = 4 × sin(π/6) = 4 × 0.5 = 2.000
Result: z ≈ 3.464 + 2i

Real-World Applications

Polar-to-rectangular conversion is essential across multiple engineering and scientific fields:

AC Electrical Engineering: Phasors representing voltage, current, and impedance are expressed in polar form. Adding or subtracting phasors — required by Kirchhoff's voltage and current laws — demands rectangular form. As demonstrated in Caltech's guide to complex numbers in circuit analysis, converting between forms is a routine task when analyzing RLC circuits.
Signal Processing: Fourier transforms yield frequency-domain components in polar form (amplitude and phase). Filtering algorithms operate in rectangular form, requiring constant conversion.
Control Systems: Transfer function analysis and Bode plot interpretation involve converting between polar gain-phase representation and rectangular real-imaginary form.
Navigation and Robotics: Range-and-bearing sensor data (polar) must convert to x-y grid coordinates (rectangular) for path planning and simultaneous localization and mapping (SLAM).

Methodology and References

The conversion formulas implemented here follow the standard treatment presented in Khan Academy's precalculus module on converting complex numbers from polar to rectangular form and the rigorous derivation provided in Paul Dawkins' Complex Number Primer at Lamar University. Additional academic grounding is drawn from the Grand Valley State University open-source textbook on complex numbers and polar coordinates. All computations use IEEE 754 double-precision arithmetic, providing approximately 15 to 17 significant digits of accuracy.

Reference

Frequently asked questions

What is the formula to convert a complex number from polar to rectangular form?

The formula is z = r(cosθ + i sinθ) = a + bi. The real part is computed as a = r cosθ and the imaginary part as b = r sinθ. For example, z = 3∠60° gives a = 3 cos(60°) = 1.5 and b = 3 sin(60°) ≈ 2.598, so the rectangular form is z ≈ 1.5 + 2.598i. The angle can be in degrees or radians.

How does entering the angle in degrees versus radians affect the conversion result?

Trigonometric functions require radian input internally. When degrees are selected, the calculator multiplies the angle by π/180 before computing. For example, 90° becomes π/2 ≈ 1.5708 rad. If 90 is entered with radians selected, cos(90 rad) ≈ −0.448 — a completely incorrect result. Always confirm the angle unit matches your input to avoid silent errors in the real and imaginary parts.

Can the real part (a) or imaginary part (b) of a complex number be negative in rectangular form?

Yes, both components can be negative depending on the quadrant the angle occupies. Angles between 90° and 270° produce a negative real part because cosine is negative in the second and third quadrants. Angles between 180° and 360° produce a negative imaginary part because sine is negative in those quadrants. For example, z = 5∠210° converts to a = 5 cos(210°) ≈ −4.330 and b = 5 sin(210°) = −2.5, giving z ≈ −4.330 − 2.5i.

What does the magnitude check output verify in the polar to rectangular calculator?

The magnitude check computes √(a² + b²) from the converted rectangular components and compares the result to the original magnitude r. A correct conversion recovers the original modulus exactly. This round-trip test — polar to rectangular and back to polar magnitude — is a standard engineering sanity check. Residual discrepancies of roughly 10⁻¹⁵ are normal IEEE 754 floating-point rounding artifacts and do not indicate a conversion error.

Where is polar-to-rectangular conversion most commonly used in electrical engineering?

AC circuit analysis is the primary application. Voltages, currents, and impedances are represented as phasors in polar form, with a magnitude (RMS amplitude) and a phase angle. Applying Kirchhoff's voltage or current law requires adding or subtracting these phasors, which is only straightforward in rectangular form. For instance, an impedance of 50∠30° Ω converts to 43.30 + 25i Ω before combining with other series or parallel impedances. This process is documented thoroughly in Caltech's AC circuits curriculum.

What is the difference between polar form and rectangular form of a complex number?

Rectangular form (a + bi) locates a complex number using its real part a and imaginary part b on a Cartesian grid. Polar form (r∠θ) locates the same point using its distance from the origin r and its angle θ from the positive real axis. Polar form simplifies multiplication and division — multiply the magnitudes, add the angles — while rectangular form simplifies addition and subtraction. Both representations describe the same point in the complex plane and are mathematically equivalent.