Curious, this may be more of a philosophical question: what is the reason why these organic-seeming movements can be modeled mathematically, specifically with trigonometric functions? Why do biological processes seem to obey trig like so?
Any motion can be modeled mathematically. There are probably infinite equations that provide "organic looking motion". I could make one that made really good looking motion but was really long and complex and you could have a parallel person asking "why is nature so messy and not giving us simple motion?". The insight here is to not attach any special significance just because "it looks organic". In reality what we are probably saying is that it "looks smooth". (In fact you could look at Bezier curves which are often used for animations to see non-trigonemtric examples, which while easy to define are relatively complex to actually run through http://en.wikipedia.org/wiki/Bézier_curve . All of iOS's built-in famous smooth animations use these.)
So let's break these down. All of these examples are periodic. So for starters we need a good periodic function. We could have started with y = |(x%2) - 1| ( http://tolmasky.com/letmeshowyou/notsmooth.png ). That would give us the back and forth motion we want, but would have not looked smooth, since we linearly increase, then linearly decrease, and in the middle abruptly change. Your eye would catch that abruptness and that's what would give it the "inorganic feel" (despite the fact that there are plenty of things in biology that are super abrupt).
So what we want instead is the same kind of back and forth, but rounding off the edges. You could imagine forming a function by taking a normal parabola (y=x^2) and an inverted one (y=-x^2), and putting them together to get the rounded bottom and top corners ( http://tolmasky.com/letmeshowyou/twoparabolas.png ).
Luckily for us, we've already discovered a class of functions that give us something similar in the form of sine and cosine. Now, the reason why these are smooth and periodic is simply because the values of these functions are derived from tracing along a circle ( http://www.galaxygoo.org/math/sineCurve.html ). Now that you know this, you have a "tool" for your mathematical bag of tricks: every time you want to do something smooth and periodic, you are probably going to reach for sine or cosine. Similarly, if you want to now tweak this and have things ramp up in speed or down, you could try using exponents, etc.
Hopefully this sheds a little light on what's going on here.
Would we then always get better approximations the higher the order?
EDIT: I ask this because linear acceleration doesn't seem like something which might readily occur in nature.
d^2 x
----- = -g
dt^2
x = -1/2 g t^2 + v_0 t x + x_0
d^2 x
----- = -x
dt^2
x = a sin t + b cos t
The chemical and physical systems in living tissue are much more complicated. Our best models for them often incorporate nonlinear elements (the sine wave example is linear) or a large number of variables. But periodic behavior is still quite common, and the sine wave is in some senses the simplest function that is smooth and periodic.
Note: The one musical instrument for which differential equations are not the best model is the digital synthesizer.
I'm not sure I would say that biological processes obey anything - merely that we can find it useful to describe real world behaviour in terms of these abstract mathematical functions. Of course, it's interesting to speculate if there is a link between the level of maths required to usefully model the real world and the level of maths we are happy working with - i.e. if the real world was much more complex would we also have developed a much more richer natural ability for mathematics.