A sinusoid of fixed angular frequency can be represented by a complex number that stores its amplitude and phase while leaving the common time dependence implicit. This complex representation is a phasor.

If

$$x(t)=X_m\cos(\omega t+\phi),$$

write

$$x(t)=\Re{X_m e^{i\phi}e^{i\omega t}}.$$

The phasor is

$$\underline X=X_m e^{i\phi}=X_m\angle\phi.$$

What the phasor omits

A phasor does not represent an arbitrary time-varying signal. It represents one sinusoidal component at a fixed frequency. The factor $e^{i\omega t}$ is common to all quantities in the same steady-state problem and is temporarily suppressed.

Phase differences

If

$$\underline V=10\angle30^\circ$$

and

$$\underline I=2\angle(-20^\circ),$$

then voltage leads current by

$$50^\circ.$$

Phasors are therefore a change of representation that turns same-frequency sinusoidal relationships into complex algebra while preserving amplitudes and phase differences.