I'm working with harmonic graphs and breaking down a given graph into the sum or product of sine and cosine functions. I'm okay on breaking them into the sum of two functions, but the product is more difficult, with one important aspect:
I can easily determine if the "larger" one (that you can draw enveloping the given harmonic curve) is sine or cosine and determine the equation for that one. I have a harder time determining whether the other function is sine or cosine.
I know if my harmonic does not start at (0,0) (assuming there's no vertical shifting going on), that both functions must be cosine (since sin(0) = 0, the product at x=0 would be 0). However, when the harmonic starts at, say, (0,5), how do I determine whether the second function is sine or cosine?
I've tried picking "test points" and then seeing if a product of cos*cos or cos*sin gives the appropriate value, but this method is pretty sketchy. I've also noticed that in the "middle" of the product wave, as the wave diminishes in amplitude, that sometimes the graph crosses over the x-axis and sometimes it just comes up and is tangent to the axis before descending again, rather than crossing it. (Or, approaches from the top, comes down and is tangent to it, and then ascends again.)
Is this observation about the graph important? Is there some "easy" way to determine whether the second function is sine or cosine?