I was toying around on Desmos, and I noticed that when successive sine functions were added $(\sin(x) + \sin(2x) + \sin(3x) + \ldots)$, it seemed to form a shape. Here is the graph of that shape.
It looks a little bit like a contorted tangent function or a logarithmic function, but I was wondering how I could find a function that exactly traced out
- the maximum values of the sine function,
- the minimum values of the sine function, and
- the average values of the peaks and valleys.
The function traced by the derivative is also quite interesting; it looks somewhat like a secant function. Any further information on this topic or directions in which to further my understanding on this topic would be greatly appreciated!
EDIT: On a second reread, I am looking more for a function that accomplishes this without the oscillations of the sine waves. In this graph the tangent function almost accomplishes this for the average case, but not quite.