What is the derivation for the sinusoidal waveform in an alternating current?

[sandy.bridge]

Hello all,
I'm trying to find the derivation for the alternating signal. For example,

let b=width of loop
a=length of loop
Hence,
ε=vLB=vaB=ωraB=\frac{1}{2}ωabB=\frac{1}{2}ωAB
I cannot seem to find the derivation of how one gets from this step, to the sinusoidal waveform. Any help is greatly appreciated.

 

[256bits]

Your sin comes from the fact that in the up and down direction V has a magnitude that varies sinusoudally. ie At the top and bottom, 0 deg and 180 deg, if y is the vertical direction , Vy=0. At 90 deg and 270 deg Vy has its maximum value wr.

 

[sandy.bridge]

So, essentially the first part of the derivation has nothing to do with the rest? My notes jump from the first part that I depicted directly to the sinusoid function, without any intermediate steps.