Trigonometry: Unit Circle or Right Triangles

After a conversation with a colleague yesterday, I started doing some serious thinking.  In the several semesters that I have taught Trigonometry, which I think totals about 5 or 6 times, I have been using our adopted textbook which takes the unit circle approach to introduce students to trigonometric functions.  In other words, the functions are defined in terms of the coordinates of points on the outside fo the unit circle where an angle subtends the circle.

Now, I have varied when the right triangle approach is introduced, but it has always been after the unit circle definition.  In the current version of the text, it isn’t until the following chapter that the old familiar SOHCAHTOA (sine is opposite over hypotenuse, etc.) is even brought up.  Every time I teach Trig., I feel uncomfortable waiting this long, especially since, when I am calculating the value of these functions in my head, I am picturing right triangles.  Perhaps it is simply because it is the way I taught, but I have always been better able to make the students grasp how to calculate the trig. functions this way. 

Of course, as a mathematician who has used trig functions in all sorts of higher level courses, such as Differential Equations or Analysis, I see the validity in conveying to students the functional nature of the trig functions which is something they see much better through the unit cricle approach.  Nevertheless, the students just don’t seem to grasp the functions as well, this way.  That has been my experience.

In response, I started doing a little literature searching this morning.  It is really the first time I seriously made an effort in looking into research in Mathematics Education.  So, if any of you readers out there know of a good place to look or can quickly find any articles on the methodology in teaching trigonemetry, I would appreciate your help.  For the rest of you, how were you taught and did you prefer a particular method.

2 thoughts on “Trigonometry: Unit Circle or Right Triangles

  1. Hi, I tutor math including Trig and Calculus and I’m wondering if you had any further thoughts on this subject? I first learned trig in 1988 and I do not believe we ever encountered the unit circle. In fact, I’m still not sure what the exact reasons for teaching the unit circle are, because, similar to what you said, in my mind, I am making reference triangles, even when quickly filling in the special values on the unit circle. Also, it seems to me that reference triangles are a more generalizable concept. But I do try to link to any trig problems that I might work, with the unit circle, because it seems that most students are most familiar with that approach.

    When researching this topic just now, this is actually the first article I came across, so I’m not sure how to find any studies on this topic either.


  2. I’m nearly a decade late to the party, but I agree with you that the right triangle approach is a bit more intuitive (especially as I always think of right triangles when evaluating trigonometric functions).

    To answer Kathleen (closer to only two years late), the main benefit that I’m aware of with the unit circle is that it allows us to think of sine and cosine as actual functions, or maybe more specifically as continuous functions. We can see the values of x and y changing continuously as we move around the circumference of the unit circle. And we can use them to graph the functions much more easily (sine = y = height of someone riding a Ferris wheel).

    I’m actually teaching trig for the first time, and I find it rather sad that I have students nearly begging me to talk about the unit circle, completely confused about right triangles because they were made to memorize the coordinates of important points on the unit circle, and they’d rather re-memorize them than learn about two triangles. We will eventually get into the unit circle definition, especially to help us visualize the graphs of the functions, but for now it’s all triangles all the time!


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