Euler's Formula Calculator

Euler's formula establishes a profound bridge between two seemingly unrelated areas of mathematics: exponential functions and trigonometry. When the exponent of e is purely imaginary — that is, when you raise e to the power iθ — the result is a complex number that lies exactly on the unit circle of the complex plane. The horizontal coordinate of that point is cos(θ) and the vertical coordinate is sin(θ), so the complex number can be written as cos(θ) + i·sin(θ). This geometric interpretation means that multiplying any complex number by e^(iθ) rotates it counterclockwise by θ radians about the origin, which is the foundation of phasor analysis in electrical engineering and Fourier analysis in signal processing.

The formula is most elegantly derived by comparing the Taylor series expansions of e^x, sin(x), and cos(x). The series for e^(ix) is 1 + ix + (ix)²/2! + (ix)³/3! + …, which, after simplifying powers of i, separates perfectly into the series for cos(x) and i·sin(x). The modulus of e^(iθ) is always exactly 1 regardless of θ, confirming that it lies on the unit circle. The argument of the resulting complex number is θ itself (modulo 2π). When θ = π, Euler's formula yields the famous identity e^(iπ) + 1 = 0, which connects the five most important constants in mathematics: e, i, π, 1, and 0.

Engineers use Euler's formula constantly to simplify alternating current (AC) circuit analysis. Instead of carrying separate sine and cosine terms through calculations, circuit elements like resistors, capacitors, and inductors are represented as complex impedances, and voltages and currents are expressed as rotating phasors of the form Ve^(iωt). Signal processing relies on Euler's formula as the backbone of the Discrete Fourier Transform, which decomposes any periodic signal into a sum of complex exponentials. In quantum mechanics, wave functions are written as complex exponentials whose argument encodes phase information. Understanding and rapidly evaluating Euler's formula is therefore a fundamental skill across engineering and physics disciplines.

Euler's formula e^(iθ) = cos(θ) + i·sin(θ) applies specifically to purely imaginary exponents. For the general complex exponent z = a + iθ, the full form is e^z = e^a · (cos(θ) + i·sin(θ)), where e^a scales the modulus away from 1. This calculator evaluates only the unit-circle form. Floating-point arithmetic introduces tiny rounding errors near exact multiples of π/2 — for example, cos(π/2) may display as a very small number like 6.12e−17 rather than exactly 0 due to IEEE 754 double-precision representation; treat any value smaller than ~1e−14 as effectively zero. Angles expressed in degrees are converted internally to radians, and the output argument is given in degrees for convenience. There is no restriction on the range of θ; values outside [0, 2π) are valid and simply represent multiple full rotations around the unit circle. This tool does not handle symbolic computation and operates purely numerically.