Discovering the Angle Sum and Difference Identities

In my Advanced Methods class at Penn’s Graduate School of Education, my students are working in groups to create shared lesson plans using an inquiry approach. For a number of reasons it can be challenging for these pre-service teachers to identify appropriate topics for student inquiry, but sometimes the brainstorming they do turns into something exciting.

And so it was recently when I conferred with pre-service teacher Andrew Laskowski about planning a lesson on trig identities. His high school students were already quite familiar with the Pythagorean identities, so his group had been thinking about how to add excitement and discovery to a lesson in which students either prove or make use of the angle sum and difference formulas. As we worked on it, we came across a diagram in Wikipedia. I’ll provide the url of the Wikipedia article at the end of this post, but please don’t peek and spoil your fun.

I’ve always had trouble remembering the angle sum and difference formulas, as I often do with formulas that I’ve memorized but haven’t truly owned. Looking at the diagram, I realized that the formulas were jumping out at me. It’s an easy construction but an elegant one, and by doing it once for myself I was convinced I’d never forget it.

Andrew’s group and I spent some time working on how to present it effectively, and I hope that some of the high school students to whom this lesson will be taught will be as excited as I was about it!

We ended up with a simple diagram and challenge. It’s dynamic, of course; drag the red points to adjust the angles.

Be sure to solve it first, preferably without the hint, and be sure to see what conclusion you can draw about the purple segments, and what conclusion you can draw about the green ones. (One of the things I love is that you can solve the whole thing using just one segment length and two angles.)

When you press the “I solved it” button, several new buttons appear, and an animation shows one order in which a student might solve the triangles. The animation finishes by extracting the purple and green segments from the figure to better demonstrate the striking conclusion. (You can use the various buttons that appear to show the animation one step at a time.)

Here’s the Wikipedia article that inspired this challenge: Angle Sum and Difference Identities.

Here’s a pdf you can download if you want to use this challenge with your own students, and here’s the sketch.

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