The full phasor is in fact $e^{i(kx-\omega t+\varphi_0)}$ and is such that, by definition, its real part is the physical quantity: $$ \hbox{Re}(e^{i(kx-\omega t+\varphi_0)}):=\cos(kx-\omega t+\varphi_0)\, . $$ By expanding $\cos(A+B)$ one can get the full linear combination of sine and cosine functions.

The phasor form is very useful since differentiation w/r to $t$ of the physical quantity is just multiplication of the phasor by $-i\omega$, and differentiation w/r to $x$ of the physical quantity is just multiplication of the phasor by $ik$.

Mathematically, the use of phasors transforms some differential equations for physical quantities into algebraic equations for the corresponding phasor forms, which are easier to manipulate than the corresponding equations in terms of sine and cosine.

Note that several equivalent phasors can be used to represent the same physical quantity: if your physical quantity is $\sin(kx-\omega t)$, possible phasors are $-i e^{i(kx-\omega t)}$ or $e^{i(kx-\omega t-\pi/2)}$.

The basic idea is to leverage the observation that derivatives of exponentials are easier to manipulate than derivatives of trig functions.

The full expression is in fact $e^{i(kx-\omega t+\varphi_0)}$ and is such that, by definition, its real part is the physical quantity: $$ \hbox{Re}(e^{i(kx-\omega t+\varphi_0)}):=\cos(kx-\omega t+\varphi_0)\, . $$ By expanding $\cos(A+B)$ one can get the full linear combination of sine and cosine functions.

The phasor form goes further and eliminates the $e^{-i\omega t}$ part and is very useful since differentiation w/r to $t$ of the physical quantity is just multiplication of the phasor by $-i\omega$, and differentiation w/r to $x$ of the physical quantity is just multiplication of the phasor by $ik$.

Mathematically, the use of phasors transforms some differential equations for physical quantities into algebraic equations for the corresponding phasor forms, which are easier to manipulate than the corresponding equations in terms of sine and cosine.

Note that several equivalent phasors can be used to represent the same physical quantity: if your physical quantity is $\sin(kx-\omega t)$, possible phasors are $-i e^{ikx}$ or $e^{i(kx-\pi/2)}$. To get the physical quantity, one then multiples the phasor by $e^{-i\omega t}$ and then take the real part, bypassing the clumsy use of sine, cosines throughout.