# Special Right Triangles and the Unit Circle

Maybe I didn’t pay enough attention as a Precalculus Student 13 years ago, but I never saw or knew about the connection between the so-called “Special Right Triangles” (45-45-90 and 30-60-90 triangles).  When I went though, we had to memorize the unit circle.  The way I see it, the only reason to memorize the unit circle might be its use in a small part of Calculus the following year, at which point all the students will have forgotten their precious memorized circle because of summer break and the hose that is Calculus at the start of the year.

Until recently, I also thought that the point of studying “Special Right Triangles” was to be able to find sides out quickly in special cases.  I didn’t realize that you don’t actually have to memorize the Unit Circle: you can figure out the standard points on the unit circle pretty quickly just using those two special right triangles!

A Separate Story

When I started teaching, someone told me about “the best math teacher they had ever seen.”  He was a crusty old guy who sat in the back of the classroom in a leather armchair and eat fish-chips, or something like that.  He would have the students go to the board and do all the problems, especially the weakest students.  Apparently it was one of these stories where the administration always gave him the students who failed the other math teachers’ classes because he could teach the students, despite the fact that this back-of-the-room teaching style was all he did.  At the time I had just come out of Education Grad School and was thinking to myself “yeah, okay, whatever.”

Well, fast-forward to, well, yesterday, and I decided to try this out because I was feeling a bit under the weather and was too weak to move much.  I have a class of 15 Precalculus students, mostly juniors, and I sat in the back of the classroom calling them up more or less randomly to do a particular task on the board.  At the end of the 50 minute period, we had this.

Sometimes I would ask them to do something as simple as “draw a circle” or “label the radius of the circle to be 1” (that one actually really frightened a girl for some reason).  Other times, I’d ask them to draw a right-triangle and then have them sit down and have the next person label the sides of the triangle.  As you can see from the image, with all the different hand-writing, every student go to go to the board at least 2 times.  If a student asked “is this right?” I would look away as if I wasn’t paying attention, and the student would usually redirect the question to the class.  I only had to ask “are we sure about this?” one or two times when the class was about to allow something that wasn’t true.

What surprised me was how quickly the students picked up on some things (for example: Quadrants II, III, and IV were just the negative values of the first quadrant) and how difficult other things were (for example: if the hypotenuse of a 30-60-90 triangle is 1, what are the lengths of the sides?).

It took the full 50 minutes, and I had to go up to the board to point out or explain how the coordinates represented the trig functions cosine and sine of the angle, but I really think the students understood the unit circle: where it comes from and why we have it.  In fact, they had seen this the year before, where the teacher called it the “Death Star”, but they didn’t even recognize it was the same thing until I sketched a picture of Alderaan blowing up at the very end of the class period.

Prerequisite Knowledge

The students had learned their Special Right Triangles the week before, and we have been going over right-triangle trigonometry for a few weeks now, so they were ready to take on this task with minimal help from me.

Student Feedback

I asked a student what she thought of doing class that way and she said that she liked it because it didn’t let her “zone out”: she had to pay attention the whole time in case she was called up.  I was very careful to quickly stop any teasing of the person at the front because I wanted the students to feel comfortable going up to the board.  I pointed out to the students that they didn’t want to be teased when it was their turn at the board, so they should be nice to the person up there now.

As we finished the activity, and I pointed out that they had “learned” this last year, they made comments like “oh, this makes so much more sense now!” and “it’s so easy now”.  Comments like that always make me feel good and I think I can safely say that every single one of the 15 students in that class could recreate the unit circle if given enough time. (We’ll find out if that claim is true when I quiz them on that soon!)

I won’t teach that way every day, but it’s definitely something I will use in the future.  I’m interested to see how the seniors (the non-honors Precalculus class) would handle me teaching like this.  I am definitely going to try it out on them, too!

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### 3 responses to “Special Right Triangles and the Unit Circle”

1. My guess is that they will remember this day – and the knowledge they represented – for a good, long time. Well done!

2. teresa

I like this idea. Could you list the order you went through as you started out? Did you have the students draw the special triangles first? Identify the 90 degree angles next? etc?

• Well, they knew about special right triangles in the days leading up to this lesson, so I guess students were thinking about these kinds of triangles already.

So I started off just by asking them to draw a circle of radius 1 with the center at the origin (this actually was somewhat difficult for them to do!). Then I just had them draw rays, one at a time, emanating from the origin at specific angle (I think we had also talked previously about angles is “standard form”). With each ray drawn, I had a separate student draw a triangle off to the side matching the triangle within the unit circle, and just said something like “oh, and label the hypotenuse 1”. They figured a lot of it out after the first one, and especially got going quickly after we finished the first quadrant.