Is VOUT/VIN Gain Valid When VIN Is a Sin/Cos Function and Can Be Zero?

Hi guys!I know what's gain and what's meaning but there's something confusing me in mathematics/equations of GAIN.GAIN=VOUT/VIN, but it may that VIN function of sin/cos then there's specific dots that's VOUT/VIN is going to infinity which means that VIN=0 over specific points in the function of sin/cos of VIN ! ... So how we actually/at all divide at all on VIN which it's zero .. so Vout/Vin is legitimate in math's rules .. 
Any help? ..in brief what's confusing me how we actually can divide by VIN which it could be function of sin/cos and sin/cos we know it could be zero in some points !! so divide by zero is legitimate in electricity?!!!!!!

You don't take the points at a specific instant.  You take the overall amplitude of the output voltage divided by the overall amplitude of the input voltage.  It doesn't matter if you use the average value, the peak value, or the RMS value, as long as you take  the same measurements of the input and output waves.

I didn't get you well sorry for that because I want to learn, I know what's that RMS  bla bla bla .. but what's confusing once again ..if VOUT/VIN then if VIN is SIN function as voltage input , then we divide Vout/Vin for all points of VIN .. which means that because Vin is sin then there's a point Vin(x0) = 0 then we are dividing Vout(x0)/0 so ..how is that legitimate? it's opposite of math, in math we can divide by zero ! 
So?!
Frankly I didn't understand your explanation "You take the overall amplitude of the output voltage divided by the overall amplitude of the input voltage" , but if overall is right, then specific is definitely right ... and by your explanation you are saying that you can divide by zero? because Vout/Vin in general so in specific we get Vout/0 then its legitimate? 

Ryan,When talking about gain in an AC circuit, first you find the RMS values of the input and output. The RMS value is always a positive number greater than zero so no need to divide by zero.For more information on RMS see this:
https://www.electronics-tutorials.ws/accircuits/rms-voltage.html

thanks !