On Friday in one of my Advanced Precalculus classes, kids were working on figuring out the double angle formulas for sine and cosine. They got and .

And then… they got stuck.

You see, I showed them two alternative forms for the double angle formula for cosine ( and ). I *showed* them these forms. And I said: figure out where they came from.

All groups in a few minutes were on yellow cups (“our progress is slowing down, but we’re not totally stuck yet”). I didn’t want to give anything away, but I didn’t have any group have a solid insight that I could have them share with others. I let things remain a bit more, no luck, so then I said: “this looks related to something we’ve seen before… a trig identity… maybe that will be helpful. Bring in something you know to open up the problem for you.” Eventually kids realized they needed to bring in some outside information (namely: ).

I was *sure* that was going to be enough. Totally certain. But after another 5 minutes of watching them struggle, I wasn’t so sure. I didn’t want to give anything more away, but I had to because we had to move forward. But what more could I give without giving the whole show away? Since many groups were trying some crazy stuff, I said: “this is a simple one or two step thing…” Why? I just wanted them to take fresh eyes and see what they could do thinking *simply*. They kept on saying I was trying to trick them, but I told them it wasn’t a trick!

And then, in the span of the next five minutes, all my groups got it.

But what was more interesting was that we had *three different ways *to do it. As kids moved on to the next set of questions (and I breathed a sigh of relief that they figured this out), I reflected on how awesome it was that they persevered and then came up with different approaches. So while they worked, I put up the three different approaches.

And with a few minutes to go at the end of class, I had everyone put everything away and I just pointed out the embarrassment of riches they came up with. And it was great to hear the audible reactions when kids who had one way saw the other ways and say things like “ooooh, I never would have thought of that!” or “that’s so clever!”

I had (have?) so many mixed feelings when I saw how difficult this question was for my kids. And I was hyperconscious about how much time we had to spend on this. But the ending made me feel like it was time well-spent.

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I teach this as part of my algebra II class, but we approach it from the side of the angle sum formula.

That in itself seems formidable, but it’s fairly straightforward if you give them this without any of the edge lengths except for the unit edge, and let them work it out.

Then I revisit it using imaginary exponents and Euler’s equation. I know those aren’t part of the Algebra II standard, but if I’m teaching both complex numbers and exponential equations in the same year, how can I *not* bring those in?

I sure wish that Euler’s equation was part of someone’s standard, because student come into my electronics course never having heard of it, but it is the simplest, most common way for dealing with sinusoids in electronics. I’d glad give up all the rest of trigonometry for it.

INORITE!?!

It seems like any class that covers exponents, complex numbers, higher order polynomials, sequences and series should at least show off the Tayler series for exponents, then have the kids figure out what happens when you throw an i in front of the x, and then let them spend some time on Desmos seeing what the real and imaginary parts end up being. The smart ones already notice that it nicely splits up into even and odd polynomials before they even start writing the functions.

It’s so rich – it ties so much stuff together. It’s not often you get a payoff like that.