Phasor

A complex constant that represents the amplitude and phase of a sinusoid at a fixed angular frequency ( $ω$ ).

Y = A e^{j ϕ} = A ∠ ϕ

By convention, phasors are defined relative to a cosine reference.

Convert between cosine and sine.

$\sin (x) = \cos (x - 90^{o}) = \cos (x - π / 2)$

Conversions

Form	Formula
Time-Domain to Phasor	$y (t) = A \cos (ω t + ϕ) \to Y = A e^{j ϕ}$
Phasor to Time-Domain	$y (t) = Re {Y e^{j ω ϕ}} = Re {A e^{j (ω t + ϕ)}}$
Rectangular with Euler’s Formula	$y (t) = Re (Y) \cos (ω t) - Im (Y) \sin (ω t)$