How the Complex Number to Trigonometric Form Calculator works
Every complex number z = a + bi can be plotted as a point on the complex plane, with the real part a on the horizontal axis and the imaginary part b on the vertical axis. Rectangular form (a + bi) describes that point using horizontal and vertical coordinates. Trigonometric form — also called polar form — describes the same point using a distance from the origin and an angle, which is often more convenient for multiplication, division, powers, and roots.
Formula and method
The trigonometric form is z = r(cos θ + i sin θ), where:
- Modulus: r = √(a² + b²) — this is the length of the segment from the origin to the point (a, b), found with the Pythagorean theorem.
- Argument: θ = atan2(b, a) — the angle measured counterclockwise from the positive real axis to that segment.
The calculator uses atan2(b, a) rather than a plain arctan(b/a) because atan2 examines the sign of a and b separately and returns the correct angle in every quadrant, typically reported in the principal range (−180°, 180°] or (−π, π] radians. Once r and θ are known, a = r cos θ and b = r sin θ recover the original rectangular coordinates — a useful way to check the conversion.
Common sources of error
- Using arctan(b/a) directly: a calculator's ordinary arctan only returns angles between −90° and 90°, so it gives the wrong quadrant whenever a is negative. atan2(b, a) avoids this.
- Mixing degrees and radians: keep track of which unit θ is in — 53.13° and 0.9273 radians describe the same angle, but they are not interchangeable numbers.
- Dropping the "i": the imaginary term is i sin θ, not sin θ — losing the i turns a complex quantity into a real one.
- Confusing r with a or b: the modulus r is a single non-negative number describing distance, never negative and generally not equal to either a or b alone.
Checking your result
A converted value should always satisfy r ≥ 0, since r is a distance. As a quick check, r² should equal a² + b² exactly, and reconstructing r cos θ and r sin θ should return you to the original a and b (within rounding). The angle should also match the quadrant of the point: positive a and b give an angle between 0° and 90°, negative a with positive b gives 90° to 180°, and so on.
Applications
Trigonometric form makes multiplication and division of complex numbers simpler: multiply two numbers by multiplying their moduli and adding their arguments, or divide by dividing moduli and subtracting arguments. It is also the basis for De Moivre's theorem, which raises complex numbers to a power by raising r to that power and multiplying θ by it. Engineers use this form for AC circuit impedance and phasors, and it underlies rotations in signal processing and computer graphics.