Discussion from a reader:
I’m attempting to understand the sine and have read your excellent information on this subject. I was lying in bed last night trying to understand the sine in terms of something moving back and forth and realized my breathing might be an excellent example. The breath-in (in terms of air taken into the lungs) definitely slows as the lungs are filled (sine ?) and slows once again as the end of the exhale (cosine ?) approaches. Is this crazy? Can the sine be described as the instantaneous change in rate along the path of the wave?
Thanks for the note – awesome question! The natural sway we feel when breathing (as we’re really full we get less and less) definitely seems like a sine wave pattern. Actually, I just did a quick google, and it looks like someone else thinks so too:
http://www.tiem.utk.edu/~gross/bioed/webmodules/lungcapacity.html
Great intuition there.
For the pattern, sine is interesting: it can be considered the pattern itself (how much air you have in your lungs). If we look at the change (how much air you are taking in, not how much you have), it would be cosine. If we look at the change in cosine (how the air you’re taking in is diminishing) it’s negative sine.
Sine has the property of being “self-similar” in that the amount it’s changing is essentially based on a sine wave as well (and the amount that changes is based on a sine wave, and so on). In Calculus there are formal descriptions of this, but that intuition is also correct: sine’s changes look like a modified sine wave too.