How this instrument works
Trigonometric (polar) form describes a complex number by its modulus, the distance from the origin, and its argument, the angle from the positive real axis — an alternative to rectangular form's plain real and imaginary parts. Converting from rectangular to trigonometric form is the exact inverse of the companion conversion elsewhere on this site: the modulus comes from the Pythagorean theorem, r = √(real² + imaginary²), and the argument comes from the two-argument arctangent, θ = atan2(imaginary, real).
Using atan2 rather than a plain arctangent matters here for the same reason it matters for any 2D direction: a plain arctan(imaginary ⁄ real) can't distinguish a complex number from its exact opposite, since both give the identical ratio. atan2 reads the sign of each part separately and places the resulting angle in the correct quadrant automatically.
Once in trigonometric form, r(cos θ + i sin θ), multiplying or dividing two complex numbers becomes far simpler than in rectangular form: multiply (or divide) the moduli and add (or subtract) the arguments — no distributing terms across a binomial multiplication needed, which is exactly why this conversion is worth having available in both directions.
- Enter the complex number's real part into the Real part field.
- Enter its imaginary part into the Imaginary part field.
- Read Modulus (r): the distance from the origin, computed via the Pythagorean theorem.
- Read Argument (θ): the angle from the positive real axis, computed via atan2.
Worked example — the complex number 3 + 4i
The complex number 3 + 4i has modulus r = √(3² + 4²) = √25 = 5, and argument θ = atan2(4, 3) ≈ 53.13° — a direct 3-4-5 Pythagorean triple appearing again, this time as a complex number's own distance from the origin rather than a triangle's hypotenuse.
The real number 1, written as the complex number 1 + 0i, has modulus 1 and argument 0° — sitting exactly on the positive real axis. The purely imaginary number 5i, written as 0 + 5i, has modulus 5 and argument 90°, a quarter turn straight up the imaginary axis.
Questions
How do you convert rectangular form to trigonometric form?
Find the modulus with the Pythagorean theorem, r = √(real² + imaginary²), and the argument with the two-argument arctangent, θ = atan2(imaginary, real). Together, r and θ describe the identical complex number that real + imaginary·i describes.
Why use atan2 instead of a plain arctangent for the argument?
A plain arctan(imaginary ⁄ real) only sees the ratio of the two parts, which is identical for a complex number and its exact opposite — 3+4i and −3−4i give the same ratio despite pointing in opposite directions. atan2 reads both signs separately and places the angle in the correct quadrant automatically.
What is the modulus of a complex number geometrically?
It's the distance from the origin to the point (real, imaginary) on the complex plane — exactly the same distance formula used for any two points in ordinary coordinate geometry, applied here with the origin as one of the two points.
Why is trigonometric form useful for multiplication?
Multiplying two complex numbers in trigonometric form only requires multiplying their moduli and adding their arguments — far simpler than distributing terms across a binomial multiplication in rectangular form, which is why calculations involving repeated multiplication or powers often convert to this form first.
What is the argument of a purely real, positive number?
Exactly 0° — a positive real number with no imaginary part sits directly on the positive real axis, requiring no rotation at all to reach it from the reference direction.