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Complex Number to Trigonometric Form Calculator

Q: Can the magnitude be negative?

No, magnitude is always non-negative by definition. It represents a distance, which cannot be negative. If you calculate √(a² + b²), the result is either positive (for non-zero complex numbers) or zero (only if both a and b are zero). Negative values appearing in formulas represent something else—typically the argument or a coefficient, not the magnitude itself.

The trigonometric representation expresses any complex number using its distance from the origin and its angle on the complex plane. Engineers, physicists, and mathematicians use this form to simplify multiplication, division, and power operations. Enter the real and imaginary components of your complex number to obtain both the magnitude and argument, displayed in radians, degrees, and multiples of π.

Last updated: April 20, 2026

Creators Mateusz Mucha, BEng

Reviewers Anna Szczepanek and Jasmine J. Mah

4,791 people find this calculator helpful

Understanding Trigonometric Form

A complex number z can be expressed as z = r[cos(φ) + i sin(φ)], where r represents the modulus (distance from the origin to the point in the complex plane) and φ represents the argument (the angle measured counterclockwise from the positive real axis).

This form is particularly powerful because it transforms addition into geometric combination and multiplication into simple operations on magnitude and angle. When you work with complex numbers in trigonometric form, operations that appear cumbersome in rectangular notation become elegant and intuitive.

The connection between rectangular and trigonometric forms reveals the underlying geometry of complex arithmetic. The real part a and imaginary part b determine where the point lies on the plane, while the magnitude and argument describe the same location using polar coordinates.

Conversion Formulas

To convert from rectangular form z = a + bi to trigonometric form, calculate the magnitude and argument using these formulas:

r = √(a² + b²)

φ = atan2(b, a)

a — Real part of the complex number
b — Imaginary part of the complex number
r — Magnitude (modulus) of the complex number
φ — Argument (phase angle) of the complex number

Step-by-Step Conversion Process

Converting a complex number requires two calculations and careful attention to quadrant placement:

Calculate the magnitude: Apply the Pythagorean theorem to find how far the point lies from the origin. This is always a non-negative real number.
Calculate the argument: Use the arctangent function (specifically atan2) to determine the angle. The atan2 function automatically accounts for which quadrant the complex number occupies, eliminating ambiguity.
Express the result: Write the trigonometric form as r times the cosine-plus-i-sine expression, or equivalently as r times e to the power of iφ (Euler's form).

The atan2(b, a) function is superior to simple atan(b/a) because it returns angles in the correct quadrant without requiring manual adjustment. A complex number in the second quadrant (negative real, positive imaginary) produces an argument between π/2 and π, which atan2 handles automatically.

Common Pitfalls and Practical Considerations

Several mistakes frequently occur when converting complex numbers to trigonometric form.

Quadrant errors with simple arctangent — Using <code>atan(b/a)</code> instead of <code>atan2(b, a)</code> can place your answer in the wrong quadrant. The number −1 + i should yield argument 3π/4, not −π/4. Always use <code>atan2</code> to account for the signs of both components.
Forgetting to validate your result — After conversion, verify your work by computing a = r cos(φ) and b = r sin(φ). These should match your original rectangular coefficients within rounding error. This sanity check catches algebraic mistakes before they propagate.
Angle representation ambiguity — Angles repeat every 2π radians. The argument 0.5 rad and 0.5 + 2π rad represent the same direction. Most conventions restrict arguments to (−π, π] or [0, 2π). Check whether your context requires a specific range.
Pure real and imaginary numbers — Real numbers have argument 0 (or π for negative reals), and pure imaginary numbers have argument π/2 or 3π/2. These edge cases sometimes confuse students because there is no conventional arctangent value for division by zero.

Practical Example: Converting 1 + i

Consider converting z = 1 + i to trigonometric form.

First, compute the magnitude: r = √(1² + 1²) = √2 ≈ 1.414.

Next, determine the argument: φ = atan2(1, 1) = π/4 radians = 45°.

The trigonometric form is therefore z = √2 [cos(π/4) + i sin(π/4)].

You can verify this: √2 · cos(π/4) = √2 · (1/√2) = 1 ✓, and √2 · sin(π/4) = √2 · (1/√2) = 1 ✓. The Euler form would be √2 e^(iπ/4).

Frequently Asked Questions

What does magnitude mean in trigonometric form?

Magnitude (also called modulus) measures the distance from the origin to your complex number's point on the plane. For z = a + bi, it equals √(a² + b²). This value is always non-negative and determines the 'size' of the complex number. Multiplying two complex numbers multiplies their magnitudes, which is why trigonometric form simplifies power and multiplication problems.

Why does the trigonometric form matter?

Trigonometric form reveals the geometric structure of complex numbers and makes certain operations vastly simpler. Multiplying two numbers in trigonometric form means multiplying their magnitudes and adding their angles—far easier than expanding (a + bi)(c + di). Similarly, computing powers and roots becomes straightforward using De Moivre's theorem. Engineers and physicists rely on this form for AC circuit analysis, signal processing, and quantum mechanics.

How do I convert from trigonometric form back to rectangular?

If you have r [cos(φ) + i sin(φ)], compute the real part as a = r cos(φ) and the imaginary part as b = r sin(φ). For example, 2[cos(π/3) + i sin(π/3)] becomes a = 2 cos(π/3) = 2 · 0.5 = 1 and b = 2 sin(π/3) = 2 · (√3/2) = √3, so the rectangular form is 1 + i√3.

What is the relationship between trigonometric form and Euler's formula?

Euler's formula states that e^(iφ) = cos(φ) + i sin(φ). Therefore, the trigonometric form r[cos(φ) + i sin(φ)] can be written as re^(iφ), called the exponential or polar form. All three representations—rectangular (a + bi), trigonometric (r[cos(φ) + i sin(φ)]), and exponential (re^(iφ))—describe the same complex number. The exponential form is especially elegant for advanced calculations.

What happens if my complex number is a negative real number?

A negative real number like −5 has zero imaginary part, so b = 0 and a < 0. The magnitude is r = |a| = 5. The argument is φ = π (or 180°) because the point lies on the negative real axis. The trigonometric form becomes 5[cos(π) + i sin(π)]. Verify: 5 cos(π) = 5 · (−1) = −5 ✓.

Can the magnitude be negative?