Phasor

A phasor is a clever shorthand for a sine wave: instead of dragging around cos(ωt + φ), you capture the whole oscillation in a single rotating arrow described by just its length (amplitude) and starting angle (phase). Picture a spoke on a spinning wheel — its shadow on the ground traces a perfect sine wave, so the spoke's length and angle tell you everything, and you can ignore the spinning for a moment.

Because every signal in a linear AC circuit oscillates at the same frequency, you can freeze the rotation and represent each voltage and current as a stationary complex number (magnitude ∠ angle). Then the calculus of differential equations collapses into ordinary algebra with complex numbers: derivatives become multiplication by jω, and capacitors and inductors get simple impedances. Phasors are the trick that makes steady-state AC analysis no harder than DC — you solve with Ohm's and Kirchhoff's laws, then convert back to a time wave at the end.