Are AC waves actually a sin function or just appear in their alternating nature as sinusoidal waves? I mean do we use the graph of a sin wave just as a representation of how the current varies, or are the actual instantaneous values of current really an output of a sin function of the angle the generator's coil makes with the field?
 A: They are actually sine waves. You can see this, for instance, if you put an oscilloscope across the mains (disclaimer: don't do this unless you know what you are doing and you know the scope and probe are rated to hundreds of volts). There is usually quite a lot of distortion and noise on the waveform which arises from both the behaviour of the generator and transformers but also various switchy things which spit out noise (dimmer switches are notoriously noisy or used to be).
A: The electromotive force generated in the power plant as a sinusoidal pattern with frequency of 50 or 60 Hz. Generally in the generator, the rotating motion of the magnetic rotor leads to sinusoidal variation of EMF in the winding of the stator and consequently in any circuit connected to the it.
The EMF induced in the coil at any instant of time depends upon the rate or speed at which the coil cuts the lines of magnetic flux between the poles and this is dependent upon the angle of rotation, θ of the generating device. Because an AC waveform is constantly changing its value or amplitude, the waveform at any instant in time will have a different value from its next instant in time.


e is potential difference; v is voltage ; $V_m$ is max voltage 
$$Instantaneous\ voltage= V_{max} \sin\theta $$
As its a Sine curve, it also makes mathematical calculations easy.But keep in mind that its sinusoidal due to the way its generated at power plant.
