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?

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    $\begingroup$ Please identify texts which address $2$ differently than $4$, and actually, examples of at least two new texts following $1-4$, and two texts from $20$ or more years ago which follow the order you claim in you post. Also, identify the languages of publications to which you are referring (for older and newer texts). Else, it is difficult to know the context from which your points of reference from which your claims. $\endgroup$ – Namaste Sep 12 at 20:42
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    $\begingroup$ 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. $\endgroup$ – Sue VanHattum Sep 13 at 1:02
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    $\begingroup$ 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.) $\endgroup$ – Joel Reyes Noche Sep 13 at 1:38
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    $\begingroup$ 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. $\endgroup$ – Matthew Daly Sep 14 at 8:13

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:


  • $\begingroup$ Excellent video, looking forward to more of this! $\endgroup$ – Maesumi Sep 13 at 14:55
  • $\begingroup$ 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. $\endgroup$ – Benjamin Dickman Sep 14 at 21:26

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.

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    $\begingroup$ Thank you for the references. Will read with interest. $\endgroup$ – Maesumi Sep 14 at 16:36
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    $\begingroup$ @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! $\endgroup$ – Benjamin Dickman Sep 14 at 16:45
  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.


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