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Update to Trig. Expansions

posted Jun 2, 2009, 4:36 PM by Eddie Woo   [ updated Jun 2, 2009, 7:23 PM ]
Worksheets evolve. At least, I think they ought to: every tool that is used to assist learning can be refined, clarified, formatted better, made more interesting, extended and just generally improved.

That's certainly been the case with my worksheet on trigonometric expansions. It started life when I first saw a very elegant geometric proof for the tan(a+b) identity (which is equal to (tana + tanb)/(1 - tanatanb), if you're curious). When I was in year 11, I learnt this identity by rote, not thinking that there would be any value in understanding where it originated from. (In fact, at that stage, I didn't even realise that I possessed the necessary knowledge or conceptual framework to derive this result at all. I now know that it's well within the reach of a Year 10 trigonometry standard.)

So to see it gracefully demonstrated in very simple terms was an eye-opener for me. (It may be strange for you to know that, despite going to James Ruse and even studying a full maths degree at university, I only really started learning maths when I began to teach it. There's a real lesson in there for all people who really want to learn what they are studying. I think there's a stronger causal relationship between teaching and learning than studying and learning!) I immediately set out to convert what I'd seen into a handout that students could go through. And so the worksheet was born - and here are the stages it went through in its development:
  1. Original version.
    A solitary labelled diagram, with a particular triangle inscribed within a rectangle. I wrote up questions on the whiteboard to gently prod students towards seeing how the identity emerges.
  2. First revision. Realising that I was losing time and interest while I subjected students to the drudgery of the boring "copy down these questions" step, I added the questions onto the sheet itself.
  3. Second revision. I thought to myself, "If there is such an elegant geometric proof for the tan(a+b) identity, ought not there to be one for sin(a+b)?" So I did some digging, found what it looked like, drew up the diagram in Geometer's Sketchpad and added in similar questions.
  4. Third revision. Before long I realised that most of the work for finding the cos(a+b) identity is already done in finding the sin(a+b) identity, so I added on the necessary steps - and I now had a worksheet to derive all three of the angle addition expansions.
  5. Current version. That brings me to the latest one, just uploaded this morning. I added answers to each of the questions and also included questions that use the identities to (a) find exact values of actual angles, and (b) derive the double-angle expansions for sin, cos and tan.
All in a day's work! Who ever thought there would be so much fun in designing worksheets. (Yes, I'm fully aware of how nerdy and sadistic that statement sounds. Guilty of the first, hopefully not the second.)