How to determine divisor for composite sine wave

Question

Suppose I added two sine waves $f(x) = sin(2\pi x)$ and $g(x) = sin(2\pi x)$ together and then divided by 2 like this $\frac {f(x) + g(x)}{2}$. If I plot this I get a graph where all three sine waves have a max amplitude of 1. However, I've discovered that dividing by 2 doesn't work if, for instance, $g(x) = sin(4\pi x)$. In other words, if I want the peak amplitude of this composite wave to equal ~1, I have to divide by something like 1.76. I'm not sure why though, since I just approximated this number by eyeballing the graph. I'm wondering if there is a method to determining what the divisor should be to ensure that the max/min amplitude is fixed to a value (like 1) after addition of two (or ideally more) waves? Thanks in advance to all.

Answer 1

Find the maxima of $f(x)+g(x)$ by differentiation, and then take that as the normalization constant. Then, the max value of the added waves will always be $1$.