Mathematica(@mathemetica):e^{ix} sits on the unit circle at angle x while e^{-ix} sits at −x. Their average is the real value cos x = (e^{ix} + e^{-ix})/2. Their difference divided by 2 is the pure imaginary i sin x = (e^{ix} − e^{-ix})/2. The horizontal dashed line joining −e^{−ix} to e^{ix} lies at height sin x and is centered exactly on that imaginary point. Radio engineers apply the same identity when computing phase angles for AC waveforms in transmission-line design.

Mathematica

@mathemetica

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Mathematica@mathemetica

2026.07.26 08:47

e^{ix} sits on the unit circle at angle x while e^{-ix} sits at −x. Their average is the real value cos x = (e^{ix} + e^{-ix})/2. Their difference divided by 2 is the pure imaginary i sin x = (e^{ix} − e^{-ix})/2. The horizontal dashed line joining −e^{−ix} to e^{ix} lies at height sin x and is centered exactly on that imaginary point. Radio engineers apply the same identity when computing phase angles for AC waveforms in transmission-line design.