RMS stands for Root Mean Square, which shows what it is: the square root of the mean of the squares. Let's assume we take n samples. The value of sample x will be v(x∙T/n) where T is the period of the signal. The RMS value is the square root of the average of the square of all samples (x ranging from 0 thru n-1):
Next, we replace xT/n by t: t = xT/n. This means we also have to change the range of the sum sign: if x=0, t=0; if x=n-1, t=(n-1)T/n = T when n is very large. The calculation becomes more accurate if n nears infinite:
Multiply both the numerator and the denominator by the time step between each sample (which nears zero when n nears infinite), and we get the following equation:
In case of a sinusoidal signal, v(t) = A sin(2∙π∙f∙t) where A is the amplitude of the signal.
v2(t) = A2sin2(2∙π∙f∙t). You may have learned in high school that sin2(x) = 0.5(1-cos(2x)). So: