# Pedagogical considerations behind current order of presentation of trigonometry

A pre-calculus book (Precalculus ed 1 By Miller and Gerken), presents trigonometry in the following order:

1- Angles

2- Trigonometric functions defined on the unit circle

3- Right triangle trigonometry

4- Trigonometric functions of any angle

I do not see the advantage of such a presentation. Topic 2 seems less familiar than 3; and 4 is essentially same as 2. There must be a pedagogical reason for this rearrangement but I do not know what it is.

Some other texts have somewhat similar presentations. In the text by Dugopolski (4 ed) we have 1- Angles, 2- Sine and Cosine on unit circle. In the text by Neal, Gustafson, Hughes we have 1- Angles, 2- Unit circle and trigonometric functions. (That is, no mention of "sine=opposite/hypotenuse" before the unit circle is brought into the picture.)

The order I am familiar with (for example from the text by by Beecher, Penna, Bittinger) goes as follows

A- Angles

B- Right triangle trigonometry

C- Trigonometric functions of any angle (and explain uses of unit circle here)

My question is: Did the shift in order come from research? If so, what is behind it?

• One text I used to like for Trigonometry, was by Karl Smith. (This is 3rd edition. I used an older edition. amazon.com/Essentials-Trigonometry-Karl-J-Smith/dp/0534348068) It has triangle trig first. I am currently teaching Pre-calculus from Stuart's text, and the unit circle comes first. He says you can switch the order, but that has you teaching radians along with triangle trig, which is silly. I appreciate this question. Sep 13 '19 at 1:02
• I prefer the style of Paul Ryder (Plane and Spherical Trigonometry, 1942) where triangles are first taught before circles (trigonometric functions of acute angles, solution of right triangles, trigonometric functions of any angle, solution of oblique triangles, radian measure, etc.) Sep 13 '19 at 1:38
• I teach at a charter school that decided several years before my hire to teach two years of algebra and then geometry rather than the "traditional" schedule of putting the year of geometry in the middle. So when I teach right triangle trig to students this year, they'll come in knowing just the unit circle definitions. I look forward to seeing how their prior knowledge will impact them, because I'm also not yet certain that the order helps the development of the bigger picture. Sep 14 '19 at 8:13
• A possible explanation is that many of these texts are geared towards students who have taken a geometry class in the past. Hence these students may already "know" about right-triangle trigonometry. Thus trig on the unit circle is presented as the main idea, and the already known theory of right-triangle trig becomes a corollary to the new theory on the unit circle. Nov 15 '19 at 23:32

## 5 Answers

I've noticed this trend as well, and it's baffling.

The only justification I can see for it is that one of the main topics in precalculus is "functions," so they introduce the sine and cosine functions first. Then they say "Hey, guess what, these apply to triangles."

However, this strikes me as being completely backward: they should introduce the sine and cosine relations first, and then say "Hey, we can generalize these to functions." It's how I usually introduce the topic:

https://youtu.be/nyXgUWoAka0?list=PLKXdxQAT3tCt84Mw-dP8Yjlv83MwET7f1

• Excellent video, looking forward to more of this! Sep 13 '19 at 14:55
• If you find the order baffling, then I suggest reading the first (and maybe second) source that I provided in another answer. There is a lengthy history of trig with circles. Sep 14 '19 at 21:26
• The linked video is superb. Jan 4 at 7:43

I disagree with the notion that the unit circle approach preceding the triangle approach should be, if it is contextualized historically, "baffling." To this end, I suggest two pieces if you are wondering about how the unit circle gets itself into the pedagogical broaching of a subject that etymologically appears to be the study of triangles/three sided figures' measurement ["trigonometry"].

First, Bressoud's "Historical Reflections on Teaching Trigonometry" [MAA link; direct PDF link]. Citation:

Bressoud, D. M. (2010). Historical Reflections on Teaching Trigonometry. The Mathematics Teacher, 104(2), 106-112.

Second, for a longer read, there is a Ph.D. dissertation that I cited in an earlier answer:

Van Sickle, J. (2011). A History of Trigonometry Education in the United States: 1776-1900 (Doctoral dissertation, Columbia University). Link.

• Thank you for the references. Will read with interest. Sep 14 '19 at 16:36
• @Maesumi Great; I'll be interested to hear what you think. I'm sure you will see that this is not just a [recent] "trend" but rather an important part of the history of teaching trig! Sep 14 '19 at 16:45

I have been teaching trigonometry for many years and, surprisingly, did not know of this issue. I read the responses and the references therein, and I am putting my thoughts on this issue here.

The question is how to approach trigonometry, right triangles first and then unit circle; or circle first and then triangle.

I think the answer depends on the level of preparation of students, in particular how comfortable are they with the concept of functions when the section on trigonometry starts. At that point we may encounter two category of students, at one extreme are the ones who struggle with basic algebra, and at the other extreme there are students who are comfortable with a question like "Let $$f(x)=x^2+3x+7$$ for $$x \ge -3/2$$. Find $$f^{-1}$$ as well as its domain and range". Does the same book or approach work for both groups? I doubt it. I think my students will do better starting with a brief intro using the triangle approach, polar coordinates, general circle, and then the unit circle.

Just as I would not advocate teaching logarithms in precalculus as the area under a hyperbola, I will also hesitate to start with the unit circle. Unit circle is pretty and offers a clean definition but much of that neatness is lost to my students.

There are decided advantages and disadvantages of triangle approach (T) and circle approach (C). Here are some items I can recall.

1- In (T) the angle takes a back seat to the triangle. The angles are lost in the fray of so many pieces of the triangle. In (C) the focus is on the angle and the function aspect of trig entities comes to light. The student cannot miss the fact that the terminal side of the angle is rotating like the handle of a clock. The angle is prominent from the start, hence the function view becomes central.

2- In (T) you are not tied to a coordinate system, an angle need not be in the standard position, and you do not worry about an orientation for angles. But you are limited to acute angles. In (C) you get the general definition and the find the signs easily.

3- In (T) You may call your angle $$x$$. Note that this is highly confusing in the unit circle presentation. Most books even avoid writing $$y=\cos x$$, when they introduce cosine. You are essentially asking students to get accustomed to the following picture. Compare that to how students are used to seeing $$(x,y)$$ in $$y=f(x)$$.

$\cos x$">

4- In (T) no triangle side is artificially set to be 1. This makes it easier to do some basic trig identities and substitutions. Say, you want to do a trig substitution for $$\sqrt{x^2+2x+5}$$. This is less error prone for students when you avoid fractions and use a triangle with 'whole' parts.

5- In (C) you can visually approximate trig functions because you set the denominator to be 1. This takes the mystery out of trig functions. Students can eyeball a diagram and have an estimate for a trig function.

6- It is important to emphasize that the input and output of most mathematical functions are dimensionless quantities, and as such, when these functions are used in applications, their input and output are ratios of similar physical quantities. One of the rare examples of this issue that comes up in precalculus is the Richter Scale formula for earthquakes $$\dfrac{R}{R_0}=Log \dfrac{E}{E_0}$$. It would be attractive to simplify the formula by taking denominators to be 1 but then we risk leaving the wrong impression that one can take the logarithm of energy. In the same fashion, when teaching $$y=\cos x$$ we need to emphasize that both $$x$$ and $$y$$ are ratios. Otherwise students may erroneously think one can take cosine of distance, time, the physical angle itself, or any other physical quantity, and that the output is another physical quantity such as distance. This lack of familiarity with the issue of units can be seen in even advanced math students who take PDEs and evaluate some lengthy integrals related to Fourier series. If they incorrectly come to an answer such as $$L+L^3$$, where $$L$$ is supposed to be some length, they do not see the obvious unit mismatch.

1. I find it easier to think about a single number (x or y of unit circle) than a ratio.

2. Also helpful with angles greater than 180 or negative.

3. I learned it this way back in the early 80s...so it's not like some totally new fangled approach.

4. Just because it seems strange to you or you get some agreement on this message board, doesn't validate your opinion. Maybe you'd feel as I do if you'd learned it this way first. It's not like double blind trials of a drug. It's just discomfort and opinion.

It all depends on the background (i.e. the axiomatic assumtions your students are able to cope with).

I recevied, in 1973, my trigonomtry in the same order, but at the time I was already quite familar with the right triangle geomerty.

My father, who studied trigonometry for surveing purpose, not mathematics, did recived it after an introduction to computation of right and not right angle triangle. The trigonoeric circle was just a memory support to remeber the signs.

I received again trigonometry for graduation. The order was : (1) polar coordinates, (2) holomorphic functions, (3) complex exponential. I do not remember of triangle geometry but of Fourier analysis.

During my PHD, I was again exposed to trigonometry. This time the order was: (1) Lie group, (2) Linear representation, (3) Eliptic functions. I then discoved by myselft (4) the triangle geometry in not euclidian spaces which is tantamount as trigomonetry on an hyperbola, parabola or elipse (circle).

SO, I'd tend to say thjat the order does not really matter, as long as they can easily rembember the basic formulas. And, for this, in my opinion, the unit circle is the best memory support.