So, my afternoon crowd was not to be outdone by my morning crew. I slipped in a subtle reference early in the conversation with them so that they would not be inclined to simply introduce the phase shift idea. I wanted them to have a little practice untangling the mechanics involved in dealing with developing a Taylor series. They were very quick to recognize and agree that the coefficients were based on factorials so jumping from the 5th degree polynomial to the 7th degree was pretty easy for them. When I asked for the cosine they were confident about using even powers instead of odds and came to a conclusion pretty quickly. Where life got interesting was when I showed them Michael’s solution from the morning and discussed why i preferred the symmetry generated by an even powered series instead. I also discussed how Michael’s translation idea might give better results for approximating cos x with negative values of x. That’s when they stepped up and knocked me out. They suggested that we take the 6th degree polynomial approximation we had for cos x and do the following: phase shift by pi radians and reflect over the x axis. I am linking to a GeoGebra file that we created. If you want to dig into that file – here are the explanations of the functions.
a and b are self-explanatory
f is the 7th degree Taylor for sin x
g is the phase shift of this by pi/2 to approximate cos x
h is the 6th degree approximation of cos x
m is the crazy reflection/shift to move the cos x approximation backwards to another portion of the cosine curve.
Whew – what a day
This is the kind of day that makes you want to keep at it, the morning would have been enough (apparently Michael is a gifted student), but two sessions…wow. I had to laugh at the realization that cos(x+pi/2) and its relation to sin(x), I used a Geogebra sheet to help convince my Trig students of this concept…and so many we’re shocked..so it’s great to hear some students adopt this understanding inherently. I wonder if you have asked this kids why it might be a useful way to handle problems using a polynomial expansion of a function and what might be there responses.
I kind of gave it away at the beginning of the conversation by letting them know that the origins of this technique lay in pre calculator days. It’s a pretty easy sell that you can find a value when x = 0.2 for a fifth degree polynomial while it is a hopeless task to find sin (0,2) without a table or a calculator. Perhaps I should not have had that discussion up front.