A friend of mine claims to have had a bad experience with trigonometry in high school. Is this because she had a bad teacher? (Most people blame their teacher.) Or is it because she was a bad student? Or is it because trigonometry is intrinsically unpleasant? In any case, part of the issue is that people have wildly different definitions of sine and cosine. Do you remember how they were defined in your school?
In my experience, both as a student in the early ’60s and as a teacher from 1969 to the present, there are four commonly used sets of definitions. And—depending on the teacher, the student, and the definitions—the choice may result in tears, fears, or cheers. Let’s examine what’s going on, and I’ll try to keep the math to a minimum here so as to avoid losing any of my readers along the way.
First of all, some context about the opening paragraph above. When my friend and I were talking about the trig unit that used to be part of the curriculum I taught at the Crimson Summer Academy, I eventually realized that we were talking at cross-purposes because we had different definitions in mind. (You know, it’s like the difficulty in resolving a disagreement when you’re in your house and I’m in mine, and it turns out that we were arguing from different premises.) To her, a sine is a ratio between sides of a right triangle (opposite leg over hypotenuse, to be more precise). To me, a sine is the y-coordinate of a point moving around a unit circle. Derived immediately from that, it’s a periodic function. So there we have three definitions (loosely characterized here, but it’s easy enough to tighten them up to make them specific). The fourth is the infinite-series definition that I would always introduce at the end of Honors Precalculus, leading to the amazing
eiπ=–1, but that’s a digression here. All of these are complicated by whether you’re using degrees or radians.
No wonder students are confused!
Interestingly, one precalculus textbook that I have presents a choice of parallel tracks through trig: triangles before circles (the historically correct order), or circles before triangles (the “modern” way).
Of course in some ways these four definitions are all equivalent, but seeing that requires a significant amount of comfort with mathematical manipulation and abstract thinking.
To explore these issues further, read James Propp’s wonderful essay, “Unlimited Powers.” It contains considerably more math than I’ve been using here, so brew yourself a nice cup of tea before you read it.
Categories: Math, Teaching & Learning