NOT SO EASY AS PI. To bring a gorgeous Pi Day to an end in the State Line, a math teacher's lament (via Joanne Jacobs) about teaching trigonometry.

Today an old time teacher, one that I "had respected" told us that he doesn't bother teaching the trigonometric functions of the special angles. That is of the 30, 60 and 45 degree angles. He said there is no reason for the kids to know the exact values as they can figure out all multiple choice questions using a calculator, working backwards from the choices, if necessary.

That sounds like a lot of extra work, compared to knowing that sin(pi/4) = cos(pi/4) = 0.71 (to two decimal places, it thus follows that tan(pi/4) = 1), and sin(pi/6) = cos(pi/3) = 0.5. The first set of relationships is useful for setting the A triangle (a right isosceles, see example 3) in the Inland Lake Yachting Association race manual, and the second and third work to set the B and C triangles, which are 30-60-90 with the short leg to weather or to lee. And on the boat our headings are reckoned in degrees. So why did I throw in those angles as fractions of pi? I certainly wouldn't impose that on the race committee, not to mention that "pi/6 west of north" is a lot more cumbersome than "330 degrees". Math teachers, however, often impose radians on students without offering a satisfactory explanation.

The reason for this is that so many formulas become much easier to write and to understand when radians are used to measure angles.

The context from which I grabbed the quote is correct but incomplete. The real power of radians crops up in Euler's trigonometric series, and it's unfortunate that there doesn't appear to be an intuitive introduction to their use that would be suitable for ninth- or tenth-graders first encountering trig. But it's so cool to fool around with the expansions of the sine and the cosine and see all the terms cancel when you sum the squares, and it's even cooler to note that each expansion looks like part of the natural exponential, which it is: cos(pi) + i sin(pi) + 1 = 0.