Euler's identity · e^{iθ} on the unit circle
Multiplying by e^{iθ} is a pure rotation by the angle θ in the complex plane. The real part draws a cosine, the imaginary part a sine — one exponential contains both. Half a turn (θ = π) lands exactly on −1, which is why every alternating current, quantum phase and Fourier component is written this way.
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