Why use different arguments to plot a sine wave? In my Electronics class were given some examples of sine wave graphs that represent voltage in respect to time, $v(t) = A \sin(\omega t)$ and other graphs whose $x$ axis is $\omega t$ instead of $t$, like in the image below: 
I know the product $\omega t$ gives an angle, but I want to know what is the reason to use this notation instead of representing the sine wave with respect to time. Is there an advantage? Are there any examples in real life that explain the need of representing waves in this form?
 A: By using the omega, you are plotting the x axis with a dimensionless quantity which makes it easier to interpret. By using time only, you are fixing the frequency of the sine wave so the plot does not have universal application.
A: In electronics, you are frequently more interested in the phase relationship between signals because that gives rise to the most interesting effects: when voltage and current are 90° out of phase, for example, there is no net power dissipation (purely imaginary impedance).
When you look at a signal on the oscilloscope, you often tweak the time axis until you see just one or two waveforms - because you are interested more in the evolution of the wave than the time that it took. You can then decide to put some cursors on to figure out the period (or frequency) of the signal but the shape usually matters more.
With that kind of thinking, plotting a wave in terms of $\omega$ makes sense - it helps you focus on the elements of the wave that matter more. You end up plotting phase along the X axis so things like power reflection etc are immediately obvious.
That doesn't mean that plotting time is wrong - just that for electronic signals with high periodicity, it's often not particularly helpful.
A: By adding the omega, you have the zero crossings at 0, pi, 2pi, etc.  It is just a convenience to keep things oriented toward radians.
