Why does the complex representation of a voltage \$V\cdot\cos(\omega t + \phi)\$ use the cosine function?
Both sin and cos are considered sinusoidal waveforms. As a practical matter, sin and cos are essentially the same thing, just offset by 90 degrees.Since "time 0" is arbitrary, the distinction between sin and cos only matters if you are comparing phase against another signal. 

Since Euler's equation works out to cos for the real component and sin for the imaginary component, it's just more useful to use the cos than the sin for real voltage.