Complex Number to Trigonometric Form Calculator
Enter the real part (a) and imaginary part (b) of a complex number and get its trigonometric (polar) form, exponential (Euler) form, modulus, and argument in both degrees and radians. The show-your-work panel walks through every step, and a reference table covers the most common cases.
Formula
Worked example
For z = 3 + 4i: r = sqrt(9 + 16) = sqrt(25) = 5; phi = atan2(4, 3) = approx 0.9273 rad = approx 53.13 deg; trigonometric form = 5(cos 53.13 deg + i sin 53.13 deg); exponential form = 5 * e^(i * 0.9273).
What is the trigonometric form of a complex number?
Every complex number z = a + bi can be represented as a point in the Argand plane, where a is the horizontal (real) coordinate and b is the vertical (imaginary) coordinate. The trigonometric form - also called polar form - describes the same point using its distance from the origin (the modulus r) and the angle it makes with the positive real axis (the argument phi). The result is z = r(cos phi + i sin phi). This representation is particularly useful when multiplying or dividing complex numbers, raising them to powers, or finding roots, because these operations become simple arithmetic on r and phi.
How to convert rectangular to trigonometric form - step by step
Step 1 - Find the modulus: r = sqrt(a^2 + b^2). This is the straight-line distance from the origin to the point (a, b). Step 2 - Find the argument: phi = atan2(b, a). The atan2 function handles all four quadrants automatically: for a > 0 it equals arctan(b/a); for a < 0 and b >= 0 add pi; for a < 0 and b < 0 subtract pi; for a = 0 and b > 0 the angle is pi/2; for a = 0 and b < 0 it is -pi/2. The result lies in the range (-pi, pi] radians or (-180, 180] degrees. Step 3 - Substitute into the formula: z = r(cos phi + i sin phi). Step 4 - Write the exponential form using Euler's identity: z = r * e^(i*phi).
Euler's formula and the exponential form
Euler's formula states that e^(i*phi) = cos phi + i sin phi for any real phi. This remarkable identity connects the exponential function with trigonometry and means that the trigonometric form r(cos phi + i sin phi) and the exponential form r*e^(i*phi) are exactly the same thing written differently. The exponential form is especially compact and makes multiplication and exponentiation immediate: (r1 * e^(i*phi1)) * (r2 * e^(i*phi2)) = r1*r2 * e^(i*(phi1+phi2)). De Moivre's theorem follows directly: z^n = r^n * e^(i*n*phi) = r^n(cos(n*phi) + i sin(n*phi)).
Multiplying, dividing and taking powers in polar form
Once in polar form, complex number arithmetic simplifies significantly. Multiplication: multiply the moduli and add the arguments - |z1*z2| = |z1|*|z2| and phi(z1*z2) = phi1 + phi2. Division: divide the moduli and subtract the arguments - |z1/z2| = |z1|/|z2| and phi(z1/z2) = phi1 - phi2. Powers (De Moivre): z^n = r^n(cos(n*phi) + i sin(n*phi)). Roots: the k-th roots of z have modulus r^(1/k) and arguments (phi + 2*pi*j)/k for j = 0, 1, ..., k-1. Converting to polar form first and then applying these rules is almost always faster than working in rectangular form.
Common complex numbers in trigonometric form
| Rectangular z = a + bi | Modulus r | Argument phi (exact) | Trigonometric form |
|---|---|---|---|
| 1 + 0i | 1 | 0 | 1(cos 0 + i sin 0) |
| 0 + i | 1 | pi/2 | 1(cos pi/2 + i sin pi/2) |
| -1 + 0i | 1 | pi | 1(cos pi + i sin pi) |
| 0 - i | 1 | -pi/2 | 1(cos(-pi/2) + i sin(-pi/2)) |
| 1 + i | sqrt(2) | pi/4 | sqrt(2)(cos pi/4 + i sin pi/4) |
| -1 + i | sqrt(2) | 3*pi/4 | sqrt(2)(cos 3pi/4 + i sin 3pi/4) |
| -1 - i | sqrt(2) | -3*pi/4 | sqrt(2)(cos(-3pi/4) + i sin(-3pi/4)) |
| 1 - i | sqrt(2) | -pi/4 | sqrt(2)(cos(-pi/4) + i sin(-pi/4)) |
| 3 + 4i | 5 | arctan(4/3) approx 53.13 deg | 5(cos 53.13 deg + i sin 53.13 deg) |
| sqrt(3) + i | 2 | pi/6 | 2(cos pi/6 + i sin pi/6) |
Standard values - exact angles are given in pi-fraction notation.
Frequently asked questions
What is the difference between modulus and argument?
The modulus (|z| or r) is the absolute magnitude of the complex number - the distance from the origin to the point (a, b) in the Argand plane, computed as sqrt(a^2 + b^2). It is always non-negative. The argument (phi) is the angle that the line from the origin to the point makes with the positive real axis, measured counter-clockwise and expressed in radians or degrees. Together they fully describe the position of z in the plane.
Why does the argument use atan2 instead of just arctan(b/a)?
The simple formula arctan(b/a) only works when a > 0 (Quadrant I and IV). When a < 0, a regular arctan gives a result that is off by pi radians. When a = 0, division by zero occurs. The atan2(b, a) function is the two-argument version of arctangent that correctly handles all four quadrants and the axes by examining the signs of both a and b separately. This calculator always uses atan2 so you get the correct angle regardless of quadrant.
What happens when both a and b are zero?
The zero complex number z = 0 + 0i has modulus r = 0, but its argument is undefined because there is no direction from the origin to itself. The trigonometric form is simply 0. This is the only complex number without a well-defined argument.
How do I convert back from trigonometric form to rectangular form?
Given z = r(cos phi + i sin phi), simply compute a = r*cos(phi) and b = r*sin(phi). For example, z = 5(cos 53.13 deg + i sin 53.13 deg) gives a = 5*cos(53.13 deg) = 3 and b = 5*sin(53.13 deg) = 4, returning 3 + 4i. The calculator's verification row confirms this by computing r*cos(phi) and r*sin(phi) and comparing with the original a and b.
What is the principal argument and why does it matter?
Because angles repeat every 2*pi radians, there are infinitely many valid arguments for any complex number: phi, phi + 2*pi, phi - 2*pi, and so on. The principal argument Arg(z) is the unique value in the range (-pi, pi] radians (equivalently (-180, 180] degrees). Using the principal value is a mathematical convention that makes the argument a well-defined single-valued function. This calculator always returns the principal argument.
What is Euler's formula and how does it relate to the trigonometric form?
Euler's formula is e^(i*phi) = cos phi + i sin phi. It follows that z = r(cos phi + i sin phi) = r*e^(i*phi). The exponential form is a compact shorthand for the trigonometric form. The special case phi = pi gives Euler's identity e^(i*pi) + 1 = 0, often called the most beautiful equation in mathematics because it connects five fundamental constants.
How do I find the cube root of a complex number using polar form?
Convert z to polar form: z = r(cos phi + i sin phi). The three cube roots are z_k = r^(1/3)(cos((phi + 2*pi*k)/3) + i sin((phi + 2*pi*k)/3)) for k = 0, 1, 2. For example, the cube roots of 8 (= 8 + 0i, r = 8, phi = 0) have r = 8^(1/3) = 2 and angles 0, 2*pi/3, 4*pi/3, giving 2, -1 + i*sqrt(3), and -1 - i*sqrt(3).