Mathematics is always a fascinating "language" that just works. This blog posting is motivated from the recent tutoring session that I had on trigonometry. As a roboticist, trigonometry has been used regularly, and we have been taking everything for granted. However, when thinking back about how simple the idea of trigonometry is, sometimes I feel that it is really hard to believe how humans can leverage so much of such a simple idea to the sophisticated robot motion.
Trigonometry is mainly based on geometry - or, more specifically, triangles. By just defining various ratios based on the geometry of triangles, people invented 6 very special trigonometric functions. In fact, just 2 of them have been extremely useful, namely, sine and cosine. Using these 2 special functions, many geometry problems can be formulated, and "solved". When we say "solve", we have to be careful, since most of the time we don't like to solve trigo functions. But, this is beyond the scope of this blog post.
The major point that I am trying to emphasize is that geometry and algebra can be properly merged to work a lot of useful math out. The main thing is still how we can "feel" them. In order to "feel" them, I always encourage students (not to be lazy) to pick up a pen and paper, and start writing things down, especially when there is "geometry" involved. Students today have been spoiled by calculator, where they just want the formula so that they can plug in the numbers. This is not the way to "feel" math, and they just try to swallow the giant formula and just want to pass the exam. Many times, when I write things down on the paper, I can see the "aha" reaction from the student, which shows how drawings can better inspire learning and "feeling" of math.
A picture is worth a thousand words, so I think teachers should encourage that. Indeed, that is how engineers work things out, and the students should learn that from a young age. They are trained to be a "visualizer", not a "computer". A visualizer can potentially look at a problem in a big picture, but a computer can only solve a particular problem in a routine manner. Again, encourage students to visualize things, and that is always how math can be more fun!