Polar to Rectangular Complex Number Calculator

This calculator converts a complex number from polar form, a modulus r and an angle θ, into rectangular form a + bi. Enter r and the angle in degrees; you get the real part a and the imaginary part b.

Formula

a = r cos θ and b = r sin θ, so r∠θ = a + bi.

• r: modulus (length), zero or positive
• θ: angle from the positive real axis in degrees, counterclockwise positive

Worked Example

Take r = 2 and θ = 30°.

1. a = 2 × cos 30° = 2 × 0.866025 = 1.7321
2. b = 2 × sin 30° = 2 × 0.5 = 1.0000
3. So 2∠30° = 1.7321 + 1.0000i

How to read the result

Rectangular form is the one to use for adding and subtracting complex numbers: add the real parts and the imaginary parts separately. Electrical phasors written as a magnitude and phase angle convert the same way.

Limits and common mistakes

• Enter the angle in degrees. To convert radians, multiply by 180 ÷ π.
• Angles that differ by 360° give the same point.
• Small rounding errors appear at angles such as 90°, where a should be exactly 0.

Frequently Asked Questions

What is 1∠90° in rectangular form?

0 + 1i, because cos 90° = 0 and sin 90° = 1.

Does a negative angle work?

Yes. −30° gives the mirror image below the real axis: 1.7321 − 1.0000i for r = 2.

Is this the same as converting polar coordinates to x and y?

Yes. a is x and b is y.

What if r is 0?

Both parts are 0 whatever the angle.

in r | Modulus r | - | 2 | 0
in th | Angle θ | ° | 30 | -
out a | Real part a | - | r * cosd(th) | 4
out b | Imaginary part b | - | r * sind(th) | 4