Euler's formula & phasors

Euler's formula, e^{iθ} = cos θ + i sin θ, says: the exponential of an imaginary number is a point on the unit circle. Strange-looking — until you see what it does: it turns every wave into a rotating arrow, and wave arithmetic into arrow arithmetic.

Build the intuition

Why e, of all things

From calculus: e^{kt} is the function whose rate of change is k times itself. Set k = iω and the “growth” is perpetually sideways — a velocity always perpendicular to position. What moves with speed proportional to its distance, at right angles to its position? Something going in a circle. e^{iωt} isn't a trick; it's circular motion, derived.

e^{i\theta} = \cos\theta + i\sin\theta

The phasor: a wave as a rotating arrow

Picture an arrow of length A rotating ω radians per second, starting at angle φ. Its shadow on the real axis is A cos(ωt + φ) — a complete sinusoid. Amplitude = arrow length, frequency = spin rate, phase = starting angle. One arrow, three parameters, the whole wave. That arrow is called a phasor, and it's how professionals think about every oscillation.

A e^{i(\omega t + \varphi)} \;\longrightarrow\; A\cos(\omega t + \varphi)

Why arrows beat trig identities

Add two same-frequency waves with different phases: by trigonometry, a page of identities. By phasors: add two arrows tip-to-tail (you learned this in vectors!) and read off the result's length and angle. Differentiation? Multiply by iω — rotation a quarter turn and scaling. The hard operations of wave math become one-step arrow moves.

See it move

InteractiveThe circle that makes waves

Angle θ0.9

At angle 0.9 rad: height sin θ = 0.78, horizontal cos θ = 0.62. Walk the full circle and the height traces one complete sine wave — triangles and waves are the same picture.

e^{iθ} live: the point is the phasor's tip, θ is the phase you control, and the traced wave is its real-axis shadow. Spin the arrow, get the wave.

A worked example

Add two waves the phasor way

Add cos(ωt) and cos(ωt + 90°). Trig identities would take a while.
As phasors: arrow 1 points east (length 1, angle 0°); arrow 2 points north (length 1, angle 90°).
Tip-to-tail sum: length √2, angle 45° — by Pythagoras and one glance:
$\sqrt{2}\cos(\omega t + 45°)$
Vector addition from linear algebra just did your trigonometry.

Out in the world

Radio is phasor engineering

Wi-Fi and 5G encode data by setting the amplitude and phase of a carrier wave — literally placing phasor tips at chosen points (a “constellation diagram” engineers stare at daily). Each received arrow position decodes to bits. Your wireless traffic is Euler's formula, modulated.

Common confusion, cleared

“e^{iπ} + 1 = 0 is deep mysticism.”

It's the arrow picture at one instant: rotate half a turn (angle π) from 1 and you stand at −1. Beautiful, yes — but readable: the “mystical” equation is a U-turn.

“Phase is a minor detail next to amplitude and frequency.”

Phase carries alignment — it's why noise-cancelling works (opposite phase), how radio encodes data, and what makes two speakers reinforce or cancel. Many signals are unintelligible if you keep amplitudes but scramble phases.

Check yourself

PracticeQuick check

Multiplying a phasor by i does what to its wave?

Recap

e^{iθ} is the point at angle θ on the unit circle — rotation, exponentiated.
A phasor (rotating arrow) encodes amplitude, frequency, and phase at once.
Wave addition and calculus become arrow addition and rotation.

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