37

(Disclaimer: I peronally really don't care if one uses $\tau$ or $\pi$, both are just numbers for me.) But I would strongly recommend to use $\pi$. Why? In every technical literature, in many popular literature, the people always use $\pi$ (even worse: $\tau$ is used for different things than $\tau=2\pi$ which would confuse when reading those literature). ...


33

The specific identity \begin{equation}\tag{A} \tfrac{1}{1 - \sin{x}} + \tfrac{1}{1 + \sin{x}} = 2\sec^{2}{x} \end{equation} as such is probably not often encountered, but simplifications akin to \begin{equation}\tag{B} \tfrac{1}{1-t} + \tfrac{1}{1 + t} = \tfrac{2}{1 - t^{2}} \end{equation} occur frequently. For example, integration via partial fractions ...


32

I suppose my preference would be     4. The differential equations road. That is, define $x(t) = \cos t$ and $y(t) = \sin t$ to be the solutions to the following initial value problem: $$ \frac{dx}{dt} = -y,\qquad \frac{dy}{dt} = x,\qquad x(0) = 1,\qquad y(0)=0. $$ This assumes that you've proven the existence and uniqueness theorem ...


30

I feel that it is perhaps a little irresponsible to teach $\tau$ instead of $\pi$. As a first introduction, it is the norm which should be taught: teaching a rare alternative to $\pi$ only serves to confuse students, especially when almost all available resources use $\pi$ instead of $\tau$. Imagine a student's confusion when they see $\tau$ in class and $\...


29

An identity always holds for some values of the free variables. Sometimes the allowed values are all real numbers, sometimes something else. In this case the identity holds whenever $\sec x$ and $\tan x$ are defined (they are defined in the same set). In a certain limiting sense the identity is also true at $x=\pi/2$, but you probably don't want to go into ...


27

Regarding the second part of your question: Would I be doing my students a terrible disservice if I introduced them to trig functions treating them as going clockwise from 12 o'clock? I think it's important to stress that the convention is just a convention, and there is no intrinsic reason why one convention is better than another. But at the same time,...


23

One should teach $\pi$. One might discuss that there is a choice that is made that is somewhat arbitrary, and there are also reasons for a different choice but I do not see this as that relevant to make much ado about this. It should perhaps also be noticed that there are two conflicting proposals for $\tau$. Eagle (1958) proposed $\pi/2$ and Palais (2001) ...


23

Absolutely! In fact, in my opinion, the most important "math skill" that should be taught in conjunction with, and using, word problems is checking whether the answers make sense. This is an absolutely invaluable part of making any practical use of mathematics, as opposed to just blindly applying formulas for the sake of passing an exam. There are several ...


22

More of a comment than an answer but: They are all composites of more basic functions. In fact, all of the trig functions could can expressed in terms of sine, linear changes in coordinates, and rational functions. For instance: $$ \tan(\theta) = \frac{\sin(\theta)}{\sin( \frac{\pi}{2}-\theta)} $$ We certainly don't want children to have to memorize 20 ...


21

There is a simple solution to this debate:


18

Maybe draw this picture? Make it clear that the green hypothenuse is fixed in length, but the red altitude is growing and approaching that hypothenuse in length as the angle approaches $90^\circ$.          


17

$\tau$ should be taught in schools There's plenty of material arguing why $\tau$ is a much more intuitive and easier to teach concept (some of my favorites: 1,2,3) and I don't want to rehash their arguments, but if you think that $\tau$ is just to make equations look nicer, please check out those resources. The question at hand is whether math educators ...


17

The answers to a word problem should in my opinion make sense (within reasonable limits). The goal you mentioned that students should check their answers is one shared by many, as also witnessed by our recent question How to award points for sense-making at the end of a problem? Now, if it is not a given any more that the solution does indeed make sense, ...


16

My strongly held opinion is that some exact solutions are conceptually fundamental. Knowing that $\sin 45^\circ = \frac{1}{\sqrt{2}}$ (rather than approximately 0.7071) may not be important for applications in engineering — although it is certainly necessary in higher mathematics, and I wouldn't be surprised to see it in, say, theoretical physics &...


15

I cannot imagine a context where it would benefit a student to know about $\tau$ over and above knowing about $\pi$. Every text book a student will ever encounter will exclusively talk about $\pi$. Every calculator a student will ever use will have the constant $\pi$ pre-programmed in. Although there are perfectly lovely arguments to show that $\tau$ is a ...


15

Many high school geometry textbooks define an angle as simply the union of two rays with a common endpoint The advantage of this definition is its simplicity. Among its disadvantages: It does not serve well for capturing the idea of a "direction": That is, there is no way to distinguish between a clockwise and a counterclockwise rotation. It more or ...


14

As others said, degrees are taught, since they are still used. So, the question becomes why are they still used. To purely work with fractions would not be very convenient for various somewhat everyday things, since many people are more used to/better at operating with integers. So to really use $\tau$ and fractions thereof seems incovenient, and one ...


14

I feel as though a lot of this misses the point. Teaching the unit circle and trig functions in terms of "one turn" makes a lot more intuitive sense to young students than do arbitrary variables tau or pi. It just so happens that Tau is equivalent to one turn. Once students grasp the concept of the unit circle, then the pi conventions can be explored.


14

I agree this memorization is not necessary. If students understand how the trigonometric functions are defined (unit circle) and know several basic identities, everything else can be derived. I think the pythagorean identity and the sine and cosine of sum of angles are sufficient. $\sin^2 \alpha + \cos^2 \alpha = 1$ $\sin(\alpha + \beta) = \sin(\...


14

It is possible to treat degrees and radians as units, just as we treat inches and centimeters as units. Just as we can convert inches to centimeters (1 in = 2.54 cm), we can convert degrees to radians (1 degree = $\pi/180$ radians). But this point of view overlooks an important fact about radians: The radian measure of an angle is the ratio of two lengths (...


14

Due to low enrollment, my AP Calc class was filled with the students who otherwise would have taken Pre-Calc this year. So you can imagine that "How much do you really need to know to see the bigger picture in calculus?" has been on my mind lately. Here's where my thoughts have fleshed out in regards to trig so far. Periodic behavior is widespread ...


13

I disagree with your statement on the usefulness of reference angles, they have more use in the plane of the unit circle. So to answer the question specifically, yes we should still teach them and one idea on relevance is below in a Physics I problem. Trigonometric functions in a complete unit circle are not one to one so the concept of a reference angle is ...


13

As several respondents have indicated, one could choose many different trigonometric functions to serve as the basic elements in terms of which other trigonometric functions are expressed and in (ancient) history other choices were made than those standard now. What we know now that was not known to the ancients that leads us to use cosine and sine is the ...


12

This aspect is discussed in great detail in §7.2 of A Course in Calculus and Mathematical Analysis by Ghorpade and Limaye. The discussion must be accessible to students undergoing their first rigorous course in analysis. The discussion is well-motivated: the authors point out in §7.1 that one can integrate all functions of the form $x^r$ for all rational $...


12

My guess is that it comes from drawing complex numbers. Putting real numbers on the $x$-axis from left to right and imaginary numbers on the $y$-axis from bottom to top matches with the way we tend to think (in cultures based on Latin at least). Once you establish that, then the function $x\mapsto e^{2\pi ix}$ takes you anti-clockwise from the positive $x$-...


11

360 has 24 divisors, more than any smaller number (this is called a highly composite number). None of its prime factors is larger than 5 ("5-smooth"). It can be divided by every natural number between 1 and 10, with the exception of 7. No smaller number can. Plus a couple more characteristics that make it very much suited for subdivision. Thus, many ...


11

The notation exists since a long time. It was used already by Irving Stringham in 'Uniplanar Algebra (1893).' This is claimed to be the earliest use on http://jeff560.tripod.com/trigonometry.html giving Cajori vol. 2, page 133 as reference. In this book, Uniplanar Algebra, the notation is used first, as far as I can see, in chapter III (The algebra of ...


11

By their definition (using a right triangle), $\sin 90^\circ$ is undefined, since a triangle cannot have two right angles. You need the definition based on a circle ($\sin \theta = y/r$) to have a value for $\sin 90^\circ$.


11

There is no such thing as a natural sign/direction convention until long after the fact. Consider another answerer's comment that anyone using the left hand rule to construct the cross product of two vectors would be mistreated. Of course, the left hand rule is exactly the correct rule to use for the path of electrons in a magnetic field? Why? Because ...


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