Complex Number to Rectangular Form Calculator

Convert a complex number from polar form (modulus r and angle θ) to rectangular form a + bi using a = r cos θ and b = r sin θ.

Quick Facts

Formula
a = r·cos θ, b = r·sin θ
r is the modulus (distance from the origin) and θ is the angle from the positive real axis; z = a + bi.

Your Results

Calculated
Real part (a)
-
a = r cos θ
Imaginary part (b)
-
b = r sin θ
Rectangular form
-
z = a + bi
Quadrant / axis
-
Location on the complex plane

Ready

Enter the modulus r and angle θ, then press Calculate.

How Polar-to-Rectangular Conversion Works

A complex number can be written in two equivalent forms: rectangular form a + bi (a real part and an imaginary part) or polar form r∠θ (a modulus/magnitude r and an angle θ measured from the positive real axis). This calculator converts polar form into rectangular form using basic trigonometry.

The formula

Given a modulus r and an angle θ, the rectangular components are:

  • Real part: a = r × cos θ
  • Imaginary part: b = r × sin θ
  • Rectangular form: z = a + bi

This follows directly from the definition of sine and cosine on the unit circle, scaled by the distance r from the origin: the point (a, b) sits at distance r from the origin at angle θ. If θ is given in degrees, it is first converted to radians (θrad = θ × π / 180) before applying the trigonometric functions, since the sine and cosine functions used here expect radians.

Worked example

For r = 5 and θ = 36.87°: a = 5 × cos(36.87°) ≈ 5 × 0.8 = 4, and b = 5 × sin(36.87°) ≈ 5 × 0.6 = 3. So the rectangular form is 4 + 3i — the classic 3-4-5 right triangle expressed as a complex number, where 5 is the hypotenuse (modulus) and 36.87° is the angle it makes with the positive real axis.

Common sources of error

  • Degrees vs. radians: entering an angle in degrees while radians was intended (or vice versa) produces a completely different point — always confirm which unit you selected.
  • Sign of the imaginary part: a negative b simply means the point sits below the real axis, not that the calculation failed.
  • Rounding too early: round only the final a and b values, not θ or the intermediate trigonometric ratios.

Reading the result

The real part places the point left or right of the origin; the imaginary part places it up or down. Together they determine which quadrant of the complex plane the number falls in — shown alongside the rectangular form below — which is useful for sketching the number or checking a hand calculation at a glance.

Frequently Asked Questions

What is the formula to convert polar form to rectangular form?
A complex number in polar form r∠θ converts to rectangular form using a = r cos θ for the real part and b = r sin θ for the imaginary part, where r is the modulus and θ is the angle from the positive real axis. The rectangular form is then written as a + bi.
Should the angle be entered in degrees or radians?
Use whichever unit matches your source and select it before calculating. Degrees are common in engineering (for example 45°), while radians are common in pure math (for example π/4). When degrees are selected, the calculator converts internally with θrad = θdeg × π / 180 before applying sine and cosine.
What do the real and imaginary parts represent?
The real part a is the horizontal coordinate on the complex plane, and the imaginary part b is the vertical coordinate. Together a + bi locates the same point as r∠θ, just expressed in Cartesian coordinates instead of polar coordinates.
Can the modulus r be negative?
By convention the modulus r is a distance from the origin, so it should be zero or positive. A "negative r" in polar notation is equivalent to a positive r with the angle rotated by 180 degrees (r∠θ = (−r)∠(θ+180°)), so enter the equivalent positive r and adjusted angle for a standard result.

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