Euler's formula

Euler's formula states that e^{iθ} = cos θ + i sin θ, relating the complex exponential to sine and cosine for real θ.

Complex exponential: The extension of the exponential function to complex inputs, defined consistently by its power series. At imaginary inputs, its values trace the unit circle and yield this identity.
Phasor: A complex number used to represent the amplitude and phase of a sinusoidal quantity. Euler's formula converts sinusoidal signals into manipulable complex exponentials.
Imaginary unit: The number i satisfying i² = −1. Its powers cycle between real and imaginary values in the exponential series.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work shaped analysis, number theory, and mechanics. He established and popularized the identity now bearing his name.
Hyperbolic functions: Functions such as sinh and cosh defined through exponential combinations. Imaginary arguments exchange trigonometric and hyperbolic functions in related identities.
Taylor series: A representation of a function as an infinite sum of derivatives evaluated at one point. Matching the exponential, sine, and cosine series proves the identity term by term.
Fourier transform: A mathematical transform that decomposes a signal into its frequency components. Its complex exponential kernels encode the sine and cosine oscillations being measured.
Complex number: A number expressible as a + bi, where a and b are real and i² = −1. The identity equates two complex-number descriptions of the same value.
Roger Cotes: An English mathematician whose work included an early relation between logarithms and trigonometric functions. His 1714 work anticipated a logarithmic form closely related to the identity.
De Moivre's formula: The identity (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) for integer n. It follows naturally by raising Euler's exponential form to integer powers.

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