Algebra • Complex Numbers & Phasors

Polar to Rectangular Complex Number Converter

Convert complex numbers from polar form (r ∠ θ) and Euler exponential form (r e^iθ) into standard rectangular form (a + bi) with exact radical evaluation and interactive Argand projection geometry.

Reviewed by Sanjay Samanta & Academic Review Board

Last Updated: September 2026

Verified Accurate: IEEE 754 & ISO 80000-2 Standard

Polar → Rectangular Converter Trigonometric Projections & Argand Triangle

Preset:

Polar Parameters (r ∠ θ)

Modulus / Radius (r ≥ 0)

Argument Angle (θ) (°)

Euler Form: 5e^(0.9273i) Quadrant I

Standard Rectangular Form (a + bi)

3 + 4i

Exact: 3 + 4i

Real Component (a) 3 r · cos(θ)

Imaginary Component (b) 4 r · sin(θ)

Complex Conjugate (z*) 3 − 4i Reflected across Real Axis

Squared Modulus (|z|²) 25 a² + b² = r²

Polar Coordinate: 5 ∠ 53.13°

Euler Exponential: 5e^(0.9273i)

Trigonometric Expansion: 5[cos(53.13°) + i sin(53.13°)]

Multiplicative Reciprocal (1/z): 0.12 − 0.16i

Argand Vector Projection Plane Right Triangle (a, b, r)

Adjacent (Real) a = 3

Opposite (Imag) b = 4

Hypotenuse (r) r = 5

Step-by-Step Mathematical Derivation

Converting from Polar (r, θ) to Rectangular (a + bi) Coordinates

Right-Triangle Trigonometry

Step 1: Angle & Modulus

Identify modulus r and argument angle θ:

r = 5

θ = 53.13° ≈ 0.9273 rad

⇒ Quadrant I

Step 2: Real Part a = r · cos(θ)

Project hypotenuse onto horizontal real axis:

cos(53.13°) ≈ 0.6000

a = 5 × 0.6 = 3.0000

Step 3: Imaginary Part b = r · sin(θ)

Project hypotenuse onto vertical imaginary axis:

sin(53.13°) ≈ 0.8000

b = 5 × 0.8 = 4.0000

Step 4: Standard Form Assembly

Combine into rectangular format z = a + bi:

z = 3 + 4i

Conjugate: z* = 3 − 4i

Direct Answer & Overview

Verified Educational Guide

How to Convert Polar Coordinates to Rectangular Complex Form

To convert a complex number from polar form z = r ∠ θ (or r · e^(iθ)) into rectangular form z = a + bi, resolve the magnitude along the two orthogonal axes using elementary right-triangle trigonometry: compute the real horizontal part a = r · cos(θ) and the imaginary vertical part b = r · sin(θ). The resulting standard complex expression is z = a + bi.

Primary Mathematical Formula Trigonometric Cartesian Projection Model

Standard Equation

ƒ(x)

Q.E.D.

z = r angle heta = r(cos heta + isin heta) = r e^{i heta} implies a = rcos heta, quad b = rsin heta

Valid for all radii r ≥ 0 and angles θ in degrees or radians.

Exact Formula

Input Parameters

Required

Modulus / Magnitude (r ≥ 0): Vector distance from the origin on the Argand plane.

Argument Angle (θ): Angle counter-clockwise from the positive real axis in degrees or radians.

Expected Outputs

Calculated

Real Component (a): Horizontal projection a = r · cos(θ).

Imaginary Component (b): Vertical projection b = r · sin(θ).

Standard Form: z = a + bi with complex conjugate z* = a − bi.

Worked Numerical Example

Instant Verification

Convert z = 5 ∠ 53.13° into rectangular form a + bi

→ a = 5 · cos(53.13°) ≈ 3.0000, b = 5 · sin(53.13°) ≈ 4.0000

z = 3 + 4i

1. Polar vs. Rectangular Complex Form: Why Conversion is Essential

In complex analysis and electrical engineering, complex numbers are expressed in two complementary geometric languages:

Rectangular (Cartesian) Form: z = a + bi, where a = Re(z) is the real coordinate and b = Im(z) is the imaginary coordinate. This form maps directly to orthogonal coordinates (a, b) on the complex Argand plane.
Polar (Phasor) Form: z = r ∠ θ = r(cos θ + i sin θ) = r e^iθ, where r = |z| ≥ 0 is the modulus (vector length from origin) and θ = Arg(z) is the argument angle directed counter-clockwise from the positive real axis.

While polar form makes multiplication, division, and exponentiation trivial (z₁ · z₂ = r₁r₂ ∠ (θ₁ + θ₂)), polar form is incapable of direct addition or subtraction. You cannot add 5 ∠ 30° + 10 ∠ 60° by adding their moduli or angles. Vector addition obeys the parallelogram law; therefore, you must convert both numbers into rectangular components (a₁ + a₂) + (b₁ + b₂)i before adding or subtracting them.

2. Mathematical Derivation: Right-Triangle Projections & Euler's Identity

Consider a vector z on the Argand plane starting at the origin (0, 0) and terminating at point P(a, b). The segment forms the hypotenuse of a right-angled triangle with length r, while the base lies along the real axis (length a) and the altitude is parallel to the imaginary axis (length b).

Fundamental Orthogonal Decomposition:

a = r · cos(θ) and b = r · sin(θ)

⇒ z = a + bi = r cos(θ) + i r sin(θ) = r[cos(θ) + i sin(θ)]

By applying Euler's formula (e^iθ = cos θ + i sin θ), we establish the direct equivalence between exponential polar form and rectangular form:

z = r · e^iθ = r(cos θ + i sin θ) = (r cos θ) + i(r sin θ) = a + bi

3. Exact Trigonometric Values for Common Reference Angles

In academic examinations and mathematical proofs, examiners expect exact radical answers rather than truncated decimal approximations. Below is a comprehensive lookup table for standard unit-circle angles:

Angle (°)	Angle (rad)	cos(θ)	sin(θ)	Unit Form (r = 1)	Example (r = 4)
0°	0	1	0	1 + 0i	4 + 0i
30°	π/6	√3 / 2	1/2	√3/2 + 0.5i	2√3 + 2i
45°	π/4	√2 / 2	√2 / 2	√2/2 + √2/2 i	2√2 + 2√2 i
60°	π/3	1/2	√3 / 2	0.5 + √3/2 i	2 + 2√3 i
90°	π/2	0	1	0 + 1i = i	0 + 4i = 4i
120°	2π/3	−1/2	√3 / 2	−0.5 + √3/2 i	−2 + 2√3 i
135°	3π/4	−√2 / 2	√2 / 2	−√2/2 + √2/2 i	−2√2 + 2√2 i
180°	π	−1	0	−1 + 0i	−4 + 0i
240°	4π/3	−1/2	−√3 / 2	−0.5 − √3/2 i	−2 − 2√3 i
270° (−90°)	3π/2	0	−1	0 − 1i = −i	0 − 4i

4. Quadrant Signs & The “All Students Take Calculus” (ASTC) Rule

Because the modulus r ≥ 0 is strictly non-negative, the algebraic signs of the real part a and imaginary part b are determined entirely by the trigonometric signs of cos(θ) and sin(θ) across the four quadrants:

Quadrant I (0° < θ < 90°)

cos(θ) > 0 ⇒ a > 0 (Positive Real)

sin(θ) > 0 ⇒ b > 0 (Positive Imaginary)

• Standard sign: +a + bi

Quadrant II (90° < θ < 180°)

cos(θ) < 0 ⇒ a < 0 (Negative Real)

sin(θ) > 0 ⇒ b > 0 (Positive Imaginary)

• Standard sign: −a + bi

Quadrant III (180° < θ < 270°)

cos(θ) < 0 ⇒ a < 0 (Negative Real)

sin(θ) < 0 ⇒ b < 0 (Negative Imaginary)

• Standard sign: −a − bi

Quadrant IV (270° < θ < 360° or −90° < θ < 0°)

cos(θ) > 0 ⇒ a > 0 (Positive Real)

sin(θ) < 0 ⇒ b < 0 (Negative Imaginary)

• Standard sign: +a − bi

Remember the mnemonic ASTC (“All Students Take Calculus”): In Quadrant I, All functions are positive; in Quadrant II, only Sine is positive; in Quadrant III, only Tangent is positive; in Quadrant IV, only Cosine is positive.

5. Cis Notation vs. Euler's Exponential Notation

You will frequently encounter three different notations when receiving polar data in engineering, mathematics, and physics textbooks:

1. Phasor Angle Notation: r ∠ θ

Standard in electrical power systems and circuit engineering. For example, a generator voltage of 120 ∠ 30° V has magnitude 120 V and phase angle 30°. Converts to 120 cos(30°) + i 120 sin(30°) = 103.92 + 60i V.

2. Cis Notation: r cis(θ)

A historical abbreviation representing cos + i sin: r cis(θ) = r(cos θ + i sin θ). Primarily used in high school trigonometry curricula.

3. Euler Exponential Notation: r e^iθ

The gold standard in modern mathematics, quantum physics, and signal processing. Exponentiation properties apply directly: e^iθ₁ · e^iθ₂ = e^{i(θ₁ + θ₂)}.

6. Step-by-Step Manual Worked Examples

Example 1: First Quadrant Special Angle Degrees & Radicals

Convert z = 10 ∠ 30° into rectangular form a + bi.

Step 1: Identify r = 10 and θ = 30°.

Step 2: Compute real part a = 10 · cos(30°) = 10 · (√3 / 2) = 5√3 ≈ 8.6603.

Step 3: Compute imaginary part b = 10 · sin(30°) = 10 · (1/2) = 5.

⇒ Standard Rectangular Form: z = 5√3 + 5i ≈ 8.6603 + 5i

Example 2: Second Quadrant Negative Real Component Obtuse Angle

Convert z = 6 ∠ 135° into rectangular form a + bi.

Step 1: Identify r = 6 and θ = 135° (Quadrant II).

Step 2: Real part a = 6 · cos(135°) = 6 · (−√2 / 2) = −3√2 ≈ −4.2426.

Step 3: Imaginary part b = 6 · sin(135°) = 6 · (√2 / 2) = 3√2 ≈ 4.2426.

⇒ Standard Rectangular Form: z = −3√2 + 3√2 i ≈ −4.2426 + 4.2426i

Example 3: Third Quadrant in Radians Radian Mode

Convert z = 8 · e^i(4π/3) into rectangular form a + bi.

Step 1: Identify r = 8 and θ = 4π/3 rad = 240° (Quadrant III).

Step 2: Real part a = 8 · cos(4π/3) = 8 · (−1/2) = −4.

Step 3: Imaginary part b = 8 · sin(4π/3) = 8 · (−√3 / 2) = −4√3 ≈ −6.9282.

⇒ Standard Rectangular Form: z = −4 − 4√3 i ≈ −4 − 6.9282i

Example 4: Fourth Quadrant Negative Angle Conjugate Symmetry

Convert z = 2√2 ∠ (−45°) into rectangular form a + bi.

Step 1: Identify r = 2√2 ≈ 2.8284 and θ = −45° (Quadrant IV).

Step 2: Real part a = 2√2 · cos(−45°) = 2√2 · (√2 / 2) = (2 × 2) / 2 = 2.

Step 3: Imaginary part b = 2√2 · sin(−45°) = 2√2 · (−√2 / 2) = −2.

⇒ Standard Rectangular Form: z = 2 − 2i

7. Real-World Engineering Applications: AC Phasors & Impedance

The conversion from polar to rectangular form is practiced millions of times daily across modern technology:

AC Electrical Circuit Analysis

AC voltage and current sources are specified in RMS magnitude and phase: V = 230 ∠ 0° V and I = 15 ∠ −36.87° A. In order to calculate the net current at a circuit junction using Kirchhoff's Current Law (∑ I = 0), engineers must convert all polar branch currents into rectangular form I_k = a_k + b_ki before summing them algebraically.

Impedance & Power Triangles

Electrical impedance in polar form Z = |Z| ∠ θ decomposes into pure resistance R = |Z| cos(θ) and pure reactance X = |Z| sin(θ), forming Z = R + jX. Similarly, apparent power S = |S| ∠ θ splits into active real power P = |S| cos(θ) (Watts) and reactive power Q = |S| sin(θ) (VARs).

Vibrations & Wave Interference

When two sound waves, seismic waves, or electromagnetic fields superimpose, their individual amplitudes A₁, A₂ and phases φ₁, φ₂ add as vectors. Decomposing each wave into cosine (in-phase) and sine (quadrature) components reveals whether the interference is constructive or destructive.

Robotics Forward Kinematics

In multi-link robotic arms, each limb segment has length L_i and joint angle θ_i. Finding the tool tip's (X, Y) coordinates requires projecting each polar link into rectangular coordinates (X = ∑ L_i cos θ_i, Y = ∑ L_i sin θ_i).

8. Common Student Pitfalls & Calculator Traps

1. Degree vs. Radian Mode Confusion

The most prevalent student mistake is computing cos(60) when the calculator is set to Radian mode instead of Degree mode. cos(60°) = 0.5, whereas cos(60 rad) ≈ −0.9524. Always double check the mode indicator (DEG or RAD) before performing trig evaluations.

2. Forgetting Negative Signs in Quadrants II, III, and IV

If you use acute reference angles rather than the full angle θ, you must manually assign the appropriate signs. For instance, in Quadrant II, a must be negative (cos < 0). Relying on full-angle evaluation (cos 150° = −√3 / 2) automatically generates the correct algebraic signs.

3. Negative Modulus Inputs

By definition, polar radius r must be non-negative (r ≥ 0). If an expression presents −5 ∠ 30°, the negative sign represents a 180° (π rad) phase shift: −5 ∠ 30° = 5 ∠ (30° + 180°) = 5 ∠ 210°, which converts to −4.3301 − 2.5i.

Fact-Checked & Verified • Computational Accuracy Standards

Updated September 2026 • Editorial Policy

Authored By

Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By

Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

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Frequently Asked Questions

How do you convert a complex number from polar to rectangular form?

To convert polar coordinates (r, θ) or exponential form r · e^(iθ) into rectangular form a + bi, evaluate the trigonometric components: compute the real part a = r · cos(θ) and the imaginary part b = r · sin(θ). For example, if r = 5 and θ = 53.13°, then a = 5 · cos(53.13°) = 3, and b = 5 · sin(53.13°) = 4, yielding z = 3 + 4i.

What is the difference between cis(θ), e^(iθ), and a + bi?

These are three representations of the identical complex number. Rectangular form (a + bi) represents Cartesian coordinates on the complex plane. Cis notation (r cis θ) is an abbreviation for r(cos θ + i sin θ). Euler's formula establishes that e^(iθ) = cos θ + i sin θ, making r · e^(iθ) the exponential polar representation. Rectangular form is optimal for addition and subtraction, while polar/exponential form is optimal for multiplication, division, and powers.

Why do you need rectangular form to add or subtract complex numbers?

In polar form (r, θ), vectors cannot simply have their radii or angles added directly because complex addition is vector addition, which obeys the parallelogram law. To add z₁ = r₁ ∠ θ₁ and z₂ = r₂ ∠ θ₂, you must first convert both into rectangular form (a₁ + b₁i and a₂ + b₂i), compute (a₁ + a₂) + (b₁ + b₂)i, and then convert back to polar form if desired.

How do negative angles affect rectangular conversion?

Trigonometric parity identities simplify negative angles: cosine is an even function (cos(−θ) = cos θ), and sine is an odd function (sin(−θ) = −sin θ). Therefore, z = r ∠ (−θ) converts to a = r cos(θ) and b = −r sin(θ), which is simply the complex conjugate of r ∠ θ (reflected across the real horizontal axis).

What are the rectangular forms of pure real and pure imaginary polar numbers?

If θ = 0°, cos(0°) = 1 and sin(0°) = 0, so r ∠ 0° = r + 0i (positive real axis). If θ = 180° (π rad), cos(180°) = −1 and sin(180°) = 0, so r ∠ 180° = −r + 0i (negative real axis). If θ = 90° (π/2 rad), cos(90°) = 0 and sin(90°) = 1, so r ∠ 90° = 0 + ri (positive imaginary axis). If θ = 270° or −90°, cos(−90°) = 0 and sin(−90°) = −1, so r ∠ −90° = 0 − ri (negative imaginary axis).

How do you convert polar form to rectangular with radians instead of degrees?

The conversion formulas remain identical: a = r · cos(θ) and b = r · sin(θ). However, your scientific calculator must be set to Radian mode (RAD). For special angles like π/6, π/4, or π/3, use exact algebraic radicals: cos(π/3) = 1/2 and sin(π/3) = √3/2, so 2 ∠ (π/3) = 2(1/2) + 2(√3/2)i = 1 + i√3.