I'm frantically trying to get my NCTM presentation for this coming Thursday into its final form. Conveniently, I am simultaneously beginning to talk with my algebra 2/trigonometry classes about sinusoidal functions, which is basically the major point of my presentation. So I had some ideas I wanted to try out with the kids, and one was a way of making sense of the "sin θ = y/r"-type of definition.
I began, instead of with SOHCAHTOA as one normally would (what? you don't know SOHCAHTOA? To be fair, neither did I before I began teaching), but with a simple sine-wave tone generated by Audacity. The wonderful thing about Audacity is that you not only hear the sound, you also see it:
Now move forward to SOHCAHTOA, and in particular, we talked about sines of particular angles: 30 degrees, 45 degrees, 60 degrees. We then plotted them. Hm. Not much to see, really. Vaguely curved, but otherwise unremarkable. Then we used calculators (!) to determine the sines of 15 and 75 degrees. We argued from geometry why the values were plausible. When we add those to our plot, we now have a good chunk of the first quarter-period of the sine wave. Then, finally, on to sines of 0 and 90 degrees, which are impossible to visualize using right triangles, but again, by plausibility arguments, the values make sense.
Now move BACK to the sine-wave tone. And lo and behold, they see that beyond the peak at 90 degrees (and doesn't it make sense that 1 should be the value of the sine function at the peak?) the graph begins to drop. And NOW we can introduce the definition of sin θ = y/r in a much more fundamental way, where y (and x and r) make more sense when connected to a point on the terminal side of an angle of rotation. And why sine being negative is okay (really! It's where y is negative!), and why it repeats.
A lot of this, of course, is just re-shuffling of topics - nothing terribly new here. But I'm starting to believe that the way to teach trigonometry is not through triangles and the wrapping function alone, but through waves and the idea of periodicity. If I can help my students to understand that there are deeper connections between triangles, coordinates, angles, and trigonometric ratios, then perhaps those tables of lots of trig values won't seem so scary to them.
Of course, maybe none of this will work, and my students will find trigonometric functions as confusing as they ever did. But I think it's worth trying.