- The lack of a trig textbook with tau was perhaps my biggest concern. At the time that I was putting my class together, I
was not aware of any college-level trigonometry textbooks that made use of tau as the circle constant. Fortunately, there were a couple of
open source textbooks written in LaTeX, a math typesetting language that I was familiar with. I modified the pi-based Stitz-Zeager college trigonometry
textbook to a tau-based version, and used it as the text for my class.
One argument I've heard against using tau in trigonometry classes is that it would be too costly to change all the pi-based textbooks. I am much more skeptical about this claim now as I was able to "tauify" two different open source textbooks over the course of several afternoons--in my spare time. It was largely search-and-replace. Some descriptions and explanations did need to be modified, but these were all straightforward changes. I see no reason why textbook publishers, who readily move chapters, change exercise sets, and add/delete topics at the request of individual colleges and universities, would be unable to produce tau-based trigonometry books.
- I did not make a production of using tau. I just began teaching using tau and continued to use it throughout. Only later in the term did I talk with them about the tau-pi issue.
I'd say that over half the students had studied some trigonometry previously, most typically in a high school class (Algebra II with Trig?). Although they had learned with pi rather than tau,
I did not observe that they had a problem with using tau. Quite the opposite, some students remarked that using tau was different for them but that it made sense.
As both a student and an educator, I had always used pi prior to Spring 2015. I think in many ways it was more of a challenge for me to begin using tau than for the students. After all, I had had decades of experience using pi in mathematical contexts. At the beginning of the term, I caught myself (OK, many times my students caught me!) making small typos (e.g., writing or saying pi when I meant tau) or mis-associating special angles (e.g., interpreting tau/4 as 45 degrees rather than 90 because of my long experience with pi/4). Fortunately, students were forgiving. It probably took me 4-6 weeks to stop thinking "in pi" and to start thinking "in tau." Based upon my own experience, teachers who plan to use tau should expect an acclimation period for themselves. Students, not steeped in the pi tradition, did not seem to experience the same re-orientation challenge that I did. For what it's worth, once I made the mental switch to tau and became "fluent," I found doing trig with tau was easier and more enjoyable.
- Learning trigonometry still has its challenges using tau. I did not find use of tau to be a panacea for all the learning obstacles in trigonometry; instead, the benefits were more subtle. I would liken it to taking an introductory physics course that employed the English system rather than SI (metric) system of measures. You would learn physics in either format, but the base-10 calculations afforded by the SI system would make for easier/faster calculations. Because tau radians is a full revolution, writing angle measures as fractions of tau allowed me to appeal to students' geometric understanding to explain concepts (e.g., half-tau is half a revolution). It seemed to me that students made frequent use of their geometric intuition, perhaps more so than if radians had been explained in terms of pi. A tutor who had worked with my tau-using students and another group of pi-using students remarked that it seemed easier for the tau-using students "to believe" the trigonometry concepts and techniques.
- Here are some brief, specific observations about teaching trigonometry with tau that I noticed in Spring 2015:
- Converting Between Degrees and Fractions of Tau Radians Was Easier
Students typically start trigonometry familiar and comfortable with degrees, and my students in the spring were no exception. I found it easier to justify and "make the case" for radian measure using tau as a complete revolution. The equating of tau radians with 360 degrees seemed very natural, and students didn't seem to have any trouble working with conversion factors of (360 degrees)/(tau radians) or (tau radians/360 degrees). I think it was actually easier for them than the conversion factor (180 degrees)/(pi radians) as most of the students were already used to dividing up a circle in 360 degrees and thinking about fractions of 360. (Note also that fractions of the form (kτ)/36 serve as a natural bridge between degrees and radians: (11τ)/36 = 110 degrees, (32.5τ)/36 = 325 degrees.)
- Writing Infinite Solutions to Trigonometric Equations Was Very Easy to Explain and Was Well Understood
Once students identified solutions for equations like sin(θ) = 1/2 for 0 ≤ θ < τ, it was easy to talk about additional solutions as just being one (or k) complete revolution(s) from the current location: θ = τ/12 + kτ, θ = 5τ/12 + kτ. In contrast, infinite solutions with fractions of pi have an extra '2' that obfuscates the geometry, forcing students to memorize patterns like 2kπ in solutions: θ = π/6 + 2kπ, θ = 5π/6 + 2kπ.
- Easier to Identify the Quadrant of Angle
Sorting and identifying the quadrant of an angle seemed more straightforward with tau. It's more natural and computationally easier to subtract out multiples of tau rather than 2pi. Then, for angles with measures less than tau radians, comparing fractions with how close they were to tau/2 or tau was accomplished with standard fraction comparison techniques (e.g., get the same denominator, or convert the fractions to decimals.)
- Special Angles and the Pesky, Hidden Denominator for τ/6 Angles
The special angles as fractions of tau are: τ/12, τ/8, and τ/6. Angles in Quadrants II, III, and IV with references angles τ/12 and τ/8 are straightforward:
τ/12 (QI); 5τ/12 (QII); 7τ/12 (QIII); 11τ/12 (QIV); and
τ/8 (QI); 3τ/8 (QII); 5τ/8 (QIII); 7τ/8 (QIV);
but note that related fractions for τ/6 get reduced in Quadrants II and III:
τ/6 (QI); 2τ/6 = τ/3 (QII); 4τ/6 = 2τ/3 (QIII); 5τ/6 (QIV);
Students had to remember that fractions of tau with denominator of 3 contained a "hidden" reference to an angle that is a fraction of tau with denominator of 6.
- Graphing Trigonometric Functions Nicer
It was easy to appeal to students' intuition that the period for sine and cosine is tau. In order to graph trig functions, we typically divide the period into four equal lengths, and doing these computations with tau as the fundamental period was much easier than with 2pi.
- Converting Between Degrees and Fractions of Tau Radians Was Easier
Phil A. Smith
American River College
2015 June 10
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