I am an adjunct faculty member, teaching introductory precalculus at Towson University.
When it comes to explaining radian measurement, students are somewhat mystified as to its purpose. I can't remember if anyone has ever raised the question in class, but I think that the general sense is, "Why do we need radians. Why can't we just use degrees?"
For the most part, you have to content yourself with saying that radians are used in higher mathematics, such as calculus, but you cannot go on to explain why, because the answer is beyond the scope of a precalculus course.
But one meaningful thing is to state that radians offer a more efficient system for measuring angles involving more than one rotation. "Quick, how many degrees does an angle have that spins around eight-and-a-half times?" This is an example that I use when introducing the subject. Using degrees, the answer involves a clumsy multiplication of 8½ times 360 degrees — 3060 degrees. Using
, the answer is 17
-radians — much simpler.
In the end, however, even that sounds a little strange, right? Things are immediate and direct when using
-radians. In that case, 8½ rotations equals 8½
-radians. One-quarter of a rotation is
-radians. One full turn is
-radians -- and not the counterintuitive 2
-radians. And so on and so forth.
I don't want to go on too long, and I am having trouble using Tex. (I have not used it for years, relying instead on MathType.)
Tau is intuitive and direct, just the opposite of pi.
I think that at least one of you accomplished and talented mathematicians out there should write a trigonometry or precalculus textbook, and show radians only in tau, not pi.