About

Every complex number z = a + bi has a unique representation in trigonometric (polar) form: r(cos θ + i sin θ). The conversion requires computing the modulus r = √a² + b² and the argument θ = atan2(b, a). A common error is using arctan(b÷a) directly, which collapses quadrant II and III into quadrant I and IV respectively, producing wrong signs. This calculator uses the four-quadrant atan2 function to guarantee correct placement on the complex plane. The result is undefined when both a = 0 and b = 0.

Note: this tool outputs the principal argument in the range (−π, π]. For applications requiring [0, 2π) simply add 2π to negative results. Floating-point arithmetic limits precision to roughly 15 significant digits. For exact symbolic answers (e.g., π÷6), a CAS is required.

Formulas

The algebraic form of a complex number is z = a + bi, where a is the real part and b is the imaginary part. Converting to trigonometric (polar) form requires two quantities.

r = |z| = √a² + b²

θ = atan2(b, a) =

{

arctan(b÷a) if a > 0arctan(b÷a) + π if a < 0, b ≥ 0arctan(b÷a) − π if a < 0, b < 0+π÷2 if a = 0, b > 0−π÷2 if a = 0, b < 0

The trigonometric form is then written as:

z = r(cos θ + i sin θ)

By Euler's formula, the equivalent exponential form is:

z = r ⋅ e^iθ

Where a = Re(z), the real part. b = Im(z), the imaginary part. r = |z|, the modulus (distance from origin). θ = arg(z), the argument (angle from positive real axis). i is the imaginary unit where i² = −1.

Reference Data

Complex Number	Modulus r	Argument θ (rad)	Argument θ (°)	Trigonometric Form	Quadrant
1 + i	√2 ≈ 1.4142	π÷4	45°	√2(cos 45° + i sin 45°)	I
−1 + i	√2	3π÷4	135°	√2(cos 135° + i sin 135°)	II
−1 − i	√2	−3π÷4	−135°	√2(cos(−135°) + i sin(−135°))	III
1 − i	√2	−π÷4	−45°	√2(cos(−45°) + i sin(−45°))	IV
3 + 4i	5	0.9273	53.13°	5(cos 53.13° + i sin 53.13°)	I
−3 + 4i	5	2.2143	126.87°	5(cos 126.87° + i sin 126.87°)	II
0 + 5i	5	π÷2	90°	5(cos 90° + i sin 90°)	+Im axis
−5 + 0i	5	π	180°	5(cos 180° + i sin 180°)	−Re axis
0 − 5i	5	−π÷2	−90°	5(cos(−90°) + i sin(−90°))	−Im axis
5 + 0i	5	0	0°	5(cos 0° + i sin 0°)	+Re axis
2 + 2√3i	4	π÷3	60°	4(cos 60° + i sin 60°)	I
√3 + i	2	π÷6	30°	2(cos 30° + i sin 30°)	I
−7 − 24i	25	−1.8546	−106.26°	25(cos(−106.26°) + i sin(−106.26°))	III
0 + 0i	0	undefined	undefined	undefined	Origin

Frequently Asked Questions

The single-argument arctan function returns values in (−π/2, π/2), which only covers quadrants I and IV. When the real part a is negative, arctan(b/a) places the angle in the wrong quadrant. The two-argument atan2(b, a) function distinguishes all four quadrants by examining the signs of both a and b independently, returning the correct angle in (−π, π]. Additionally, atan2 handles the case a = 0 without division-by-zero errors.

The complex number z = 0 + 0i has modulus r = 0. The argument θ is mathematically undefined because the origin has no unique direction from itself. This calculator reports "undefined" for the argument and trigonometric form in this case. Some conventions arbitrarily assign θ = 0, but this is not standard.

This calculator outputs the principal argument in (−π, π]. If your application requires the argument in [0, 2π), add 2π to any negative result. For example, if θ = −3π/4 (−135°), the equivalent non-negative argument is −3π/4 + 2π = 5π/4 (225°). The trigonometric form remains mathematically identical because cos and sin are 2π-periodic.

JavaScript uses IEEE 754 double-precision (64-bit) floating point, which provides approximately 15-17 significant decimal digits. For most practical inputs this is sufficient. Precision loss becomes visible when computing arguments very close to axis boundaries (e.g., a = 1e-15, b = 1). The calculator allows adjusting display precision from 0 to 15 decimal places. For exact symbolic results like π/6 or π/4, a computer algebra system is required.

Yes. Once you have the trigonometric form r(cos θ + i sin θ), De Moivre's theorem states that z^n = r^n(cos nθ + i sin nθ). Use this calculator to find r and θ, then multiply θ by your exponent n and raise r to the power n manually. For nth roots, divide the argument by n and take the nth root of r, remembering there are exactly n distinct roots separated by 2π/n radians.

Euler's formula states that e^(iθ) = cos θ + i sin θ. Therefore the trigonometric form r(cos θ + i sin θ) is identically r·e^(iθ). The exponential form is more compact and preferred in engineering and physics (especially electrical engineering where j is used instead of i). Both forms encode the same information: modulus r and argument θ.