Trigonometric Enrichment

Check your answers, Ben, I said.

He rolled his eyes and opened his math book with great labor. There was silence for a while and then some groaning and then exasperated complaints of "What!?" as he discovered there was more work to do.

He proceeded to redo his homework. I began preparing his enrichment -- a derivation of the sum-of-angles / difference-of-angles identities for sine and cosine.

It turns out that they just teach the trigonometric identities without showing where they come from. The kids just have to memorize them as self-evident truths. That's not too shocking, I suppose. I don't remember anything different from when I was that age. But those identities are so much less mysterious (and trigonometry so much less distasteful) when you learn that they really follow directly from the Pythagorean Theorem, the Distance Formula, you know: the square of the hypotenuse is ....

So I was drawing triangles and writing equations as he was reworking his problems. And I had about five pages of diagrams and equations and was just about ready when he looked over.

What are you doing? he asked.

Preparing a lesson, I said.

He groaned.

Don't worry, I said, it won't take long.

Somewhere on page four, he stopped me.

That's not the right identity, he said. That should be a minus sign.

It should? I asked.

Yes. The sine of a minus b is ... and you should have a minus sign there instead of a plus sign.

Are you sure? I asked.

Yes, he said. And he opened his book and showed me the identities printed on the inside cover.

So let's find the mistake, I said.

So we worked backwards line by line. Page 4. Page 3. Page 2. I stepped him back thru each line in the derivation. For the first time, I think he really began to understand why showing each step really is important. And then he found it.

That should be... and he pointed at my paper.

I looked down for a second and looked up at him and said, You're right. Go ahead and work the correction back down to page 4, and see if that fixes the mistake.

I walked off to answer the ringing telephone, and when I came back he was just about finished — three lines from the bottom.

Does it work? I asked.

Two lines from the bottom. Change a plus sign to a minus. One line from the bottom. Do it again. Final line, pull the minus out of the last term.

Yes! he said.

Are you sure? I asked.

He didn't know how to interpret that.

Um... he looked at it again. Yes, it fixes it.

You're right.

And I swear, as he was packing up his backpack a few minutes later, I heard him whistling the kind of whistle that he whistles only when he's proud of some piece of homework he's finally mastered -- and relieved to have finished.

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