In Precalculus this fall, we have been looking at some unexpected and subtle consequences of the Pythagorean Theorem and of the basic definitions of the sine and cosine functions. On the surface, these are simple enough principles. However, much of successful mathematical problem solving hinges on teasing out results by recognizing basic principles hidden in apparently complex situations. Trigonometry, which is essentially the broad application of the Pythagorean Theorem affords many opportunities for expanding our problem solving skills.
For example, suppose that you run over a tack on your bicycle and ride 100 feet before stopping with a flat tire. How far off the ground will the tack be? This may not be the first thing that a cyclist thinks of on flatting, but it does provide an opportunity for understanding applications of trigonometric graphs. The question of how far one can see from the window of an airplane at 35,000 feet may seem unrelated to the bicycle problem, but we have seen that solutions to both of these problems can be found from trigonometric ideas that any Geometry student has seen, but which the novice may not recognize.
In Precalculus, we work cooperatively on problems like these, experiencing frustration and success in turn, but developing our problem solving skills all along the way.