A2 Trigonometry (Parts 2 & 3)

Yes, I’m beginning with Part 2. At some point I will try and write about my approach to sec, cosec, cot and the inverse trig functions but following on from some recent Twitter comments, it seems many of us are around the sin(A+B) and Rsin(x+a) mark right now.

My approach this year has made a much bigger deal about the motivation for these formulae and techniques. Here’s a brief run-down of the stages we went through.

Part 2: sin(A+B)

Stage 1 – Motivation

A number of statements written on the board. Students to discuss in pairs if each is true or false. Typically something like this:

1.) \sqrt{a+b}=\sqrt{a}+\sqrt{b} 2.) (a+b)^2=a^2+b^2
3.) \ln(a+b)=\ln(a)+\ln(b) 4.) e^{a+b}=e^a + e^b
5.) \frac{5}{a+b}=\frac5a+\frac5b 6.) 5(a+b)=5a+5b
7.) \sin(a+b)=\sin(a) + \sin(b) 8.) (a+b)^{-1}=a^{-1}+b^{-1}

Purpose: this gives the opportunity to air a few of the typical misconceptions that always arise again near the beginning of C3 (but hopefully not so much later in the course…) But it also emphasises that most functions cannot simply be ‘expanded’ (mathematically speaking, they are not linear).

Discussion: hopefully students will correctly identify true/false. It’s nice to make improvements to several of them, such as:

  • 2.) (a+b)^2=a^2+2ab+b^2
  • 3.) \ln(a)+\ln(b)=\ln(ab)
  • 4.) e^{a+b}=e^a e^b

Most students are wise enough to not trust number 7, and thus we have some initial motivation for wondering how to ‘expand’ sin(a+b).

Stage 2 – A Proof

I’ve seen various diagrams and constructions (Cut the Knot has several pages dedicated to them) but my current favourite is this one:

trigaplusb.JPG

The students are to label all the remaining sides in the diagram (I usually model how to proceed by using say \cos\beta as an example. The worksheet I use has all the necessary labels underneath the diagram to guide them, too. Finally, the sheet poses:

trigpart2.PNG

Note, unless I have particularly high-flying students, I don’t make much of a fuss about the construction only really working if \alpha+\beta<90^\circ.

Stage 3 – Examples and Non-Example

Of course, it’s tradition to find an exact value for \sin 75^\circ at this stage, or perhaps $\latex \cos 15^\circ$, but some calculators now provide these values directly which is a shame.

I would typically demonstrate some equation that requires one or both of the new identities, but then I think a non-example is also crucial here. Something like:

Solve \sin(x+\frac{\pi}{3}) = \frac12.

Just because we can expand, doesn’t mean we always should!

That’s more than enough for one lesson. The next would begin with this slide

trig3.PNG

and then move on to the double-angle formulae.

Part 3: Harmonic Form

For this lesson, I place perhaps an even greater emphasis on motivation. There are two initial activities that I use.

Stage 0 – The Settler

trig4.PNG

I make quite a big deal about some things being “just a C2 question” – trying to make the point that C3 isn’t such a big leap as they might think, and that they really should be able to solve C2 questions by now…

Clearly the middle equation causes the problem and they’ll be clamouring to know how to solve it!

Stage 1 – The Motivation

I then give the students a sheet that begins like this:

trig5.PNG

There are a couple more parts further down, but you get the idea. Given time to work in pairs and discuss what they are able to do, the students realise that filling in the column for f(x) is relatively straightforward (and essentially C2 work, but it gives the opportunity to get a bit of help too) whereas the column for g(x) is nigh on impossible.

It’s great to see what they then try: some will plug a few values in their calculator, using a bit of trial and error to find the domain of g. Others will try and guess what the sum (or rather, difference) of a sine and cosine graph might look like. Basically though, they’re stumped.

We use Desmos to do the big reveal. Type in f(x) to check all their answers to the left column. Then type in g(x) and. Wait. What? It’s the same function? *general surprise*

Stage 2 – Backwards and Forwards

Some then realise that they could ‘expand’ f(x) using the cos(A+B) identity and we do this on the board. The stage is then set for, effectively, the reverse of this process.

Stage 3 – So, it’s not a big deal

The whole point of harmonic form is to make very difficult questions (the right column) very easy (the left column)! They love it.

Stage 4 – Practice

All of the above, perhaps with the inclusion of just another example or question is usually plenty for the first (hour-long) lesson.

I don’t have much more to add, except this idea which occurred to me for no particular reason. I make use of the classic Solomon worksheets relatively often (maybe on per week, ish) but this is the first time I’ve annotated one before issuing it to students:

trig6.jpg

I’ll try and get a higher resolution pic at some point, but the idea was to highlight the purpose (or distinguishing feature) of each question [done in red ink] and then recommend what to complete, such as “just do one of these” [done in green ink]. I’m sure the big piece of paper and colour-copying played its part in engaging the students, but they responded to it very well: for the able ones, it gave them clear guidance about what was worthwhile to complete; for the weaker ones, they had the detail of the question emphasised and felt more secure in tackling them.

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