Clarification for "Digital Signal Processing" by Steven W. Smith related to complex representation of sinusoids
Phasors are useful for the analysis of (real-valued) linear time-invariant (LTI) systems. A phasor is a complex number
$$C=|C|e^{j\phi}=A+jB\tag{1}$$
which represents a sinusoidal signal:
$$x(t)=\textrm{Re}\big\{Ce^{j\omega t}\big\}=A\cos(\omega t)-B\sin(\omega t)=|C|\cos(\omega t+\phi)\tag{2}$$
That's already the end of the story. Eq. $(2)$ is the generally accepted definition of a phasor. The definition in the book is either a mistake or an idiosyncrasy making things unnecessarily confusing.