Thursday, April 18, 2013

Unit Circle Philosophy

This year, we decided to expand the number of trig topics we teach in Algebra 2 to include the unit circle, the graphs of sine/cosine, and modeling of periodic motion. These are topics I haven't taught in quite a while, so I am getting to take a fresh look.

Here's how I started out:

1.  Review pre-requisite skills:  Angle measures, trig ratios, special triangles.
2.  Develop the idea of a reference angle.
3.  Given a special angle, draw a triangle in the appropriate quadrant, identify its reference angle, label its sides, and find the values for sine, cosine, and tangent.
4.  Replace old special triangles with new ones where hypotenuse = 1.

5.  Cut out special triangles where the hypotenuse equals one unit. Label their sides and glue them onto a unit circle.

6.  Label the points on the circle. Use the circle to evaluate sine, cosine, and tangent for all the special angles. Look for patterns.
7.  Extend the pattern to the axis angles. Use the circle to evaluate sine, cosine, and tangent for 0, 90, 180, 270.

At this point, I feel like my students have a pretty good conceptual understanding of the unit circle. What now? This is where I am stuck.

I generally don't believe in telling students to just memorize something, but I also cringe when I see a calculus student reach for their calculator when they encounter something like sin pi/2 or tan 3pi/4. 

Don't students need to be able to recall these values later on without a circle or a table in front of them?

What is the best way to tell them to remember these?

I did some searching and I decided on a mnemonic device. 

I am sort of ashamed to be using it.

So far, it is working.

My next post will be the procedure I decided to use, but I am wondering if anyone has any thoughts on memorization and the unit circle? How do you approach this in your classroom?


  1. Really getting a feel for radian measure is part of the problem, right? I wonder if watching Vi Hart's videos on the dispute about pi versus tau would make it more interesting?

  2. Hi Sue! To be honest, we are mostly just using degrees for right now. I am more conflicted about how important it is for students to be able to recall values like cos 90 or tan 135 without using a calculator, table, or circle. Is it important? If so, what is the best way to teach them that?

  3. Personally, I'm not good at recall, so I rarely ask that of my students. Yes, they should be able to find these values without a calculator, and without notes. But I would prefer that they figure the values out rather than memorize them. Can they draw the special triangles and a circle, and use those to find any trig function of any multiple of 30 or 45 degrees? That's what I want. (To this day, having taught trig dozens of times, I still have to draw a triangle in the air to figure the value for cosine of 30 degrees.)

  4. Hi Amy,
    I agree with what Sue is saying about thinking of the special right triangles in her head (which is how my principal came to class and reconstructed the unit circle one day) but whether you memorize the values of the special right triangles, or memorize the entire unit circle (which I think would be crazy) at some point in this chain there is definitely memorizing going on.
    I kind of make a game with it in my classes and have my students be able to reconstruct the unit circle in about 3:30 or less while a Taylor Swift song is playing in the background. At first they think this is an impossible, mean task, but on the second day of this usually one or two kids can do it, and it snowballs from there. I didn't enter a grade in the rank book until kids tried it 3 times, and the grades could improve until they got a 100 as well, to encourage everyone to keep trying.
    We spend time in class talking about strategies for learning the circle, usually involving pattern recognition. Although a few kids did share some patterns they discovered for learning the radian angles as well.
    One clear benefit of learning the unit circle for my students is that they can explain why a statement like sin(x) = sin(π-x) is true.
    I'm working through a trig unit now, and have been trying to post anything decent. Maybe something would be useful to you. Here is a link to my trig posts.

  5. I agree with Sue, too. I don't have the unit circle memorized and depend on my special right triangles to find specific values.

    With that being said, I do ask my students to memorize it... or at least the first quadrant ordered pairs, which they can they use to find the rest. I only switched to that method a few years ago after the calculus teacher at my school kept mentioning she wanted the kids to be able to name the values at the drop of a hat. So now I do stuff like Jason mentioned before. We construct the circle with the special triangles (patty paper works great!) and then discuss ways to help them remember it. I also challenge them weekly to make a unit circle in less than 4 minutes. They complain at first, get happy when they "get it", and then complain at the end because they're tired of it.

    It gets to be that the kids are faster at finding the values than I am. But I'm ok with that. :)

  6. What do they have to put on the circle in that 3 to 4 minutes? I might try that. We're starting up our circle trig unit right now.

  7. I'm like Kristen and I don't have it memorized myself. I used to, but then I started using

    and not only have all my kids gotten it, but I am better with it too. This only helps with the first quadrant stuff though, and I'm thinking that's not what you're looking for. I think the other quadrants just take time to see, process, and get used to via practice.

  8. Sue,
    This post : has a link to the PDF worksheet I use in my classes. It is editable if you have Acrobat or Illustrator.
    This year I began with Riley's Unit Circle measuring activity first and also used Khan Academy's Unit Circle exercise which is a good one.:

  9. Thank you all! Your comments were helpful, and I will take a look at all of these resources. I agree that it is important for students to be able to draw a triangle and figure it out -- this is how they demonstrate their understanding, I think. This seems like a good stopping point to me, except . . . . I am also a calculus teacher. When these values come up, it would sure be convenient to have a quick & dirty way to get an answer. That is why I decided to teach the cheat hand referenced by Sam. It took a little practice, but my students are spouting off answers with speed and accuracy and that makes me happy. Will it interfere with their understanding of what they are doing? I think I will do a separate post about this. I appreciate your comments!

    Sam, Couldn't the hand cheat work for all quadrants if the student could first identify whether the reference angle is 30, 45, or 60? They would have to choose the correct sign according to the quadrant.

  10. I use the unit circle as the basis for trigonometry. I have found that using the dynamic image of a GeoGebra file gives students a better understanding. We spend time getting familiar with it and estimating triangles for different angle values. We discuss which values we have memorised, and why.

  11. In my math classes, we focus a lot on finding references angles. If I can find a reference angle to something in the first quadrant, I can use my hands to figure out the sine, cosine, and tangent of the angle. Once I know the values, I adjust the signs to make the angle fit with the right quadrant. It's a really great trick, and once it is understood, it works wonders in the trig world. :)

    This video ( is what I used to really understand this method of remembering the unit circle. It's a very handy trick indeed (no pun intended!).

  12. I teach only Calculus (at least for math) at my little school and have asked the Pre-Cal teacher to be sure students are not just familiar but _comfortable_ with the unit circle. I also stress the importance of radians: It's not nearly as brain-friendly, but I say, "Math speaks radians," and show how dividing a circle into 360 degrees is quite arbitrary, but dividing it into 2π radians is inherent to a circle. I liken it to a foreign language: If you were going to live in France for a year, you could carry a phrase book with you and use it every time you wanted to order a bagel, or you could just memorize the French phrases for "I want a bagel," and "Never mind, that's way too expensive." In the long run, you'll save time memorizing.

    I think the terminating points on the unit circle helps: You essentially need to memorize three values: 1/2, √2/2, and √3/2. Those are in ascending order, and if you draw y=1/2 (or x=1/2) through the unit circle, it's even easier to see which goes where and why. Right triangles and pythag show where those three weird values come from, and the graphs show how they are arranged relative to one another. It's pretty easy to keep in your brain from there (what's my reference angle, which of the three terminating points am I at (relative to the x- or y-axis depending on trig function).

    I love your blog, by the way.

  13. Loving all the hints that have been shared here, so wanted to share mine in case it helps someone with radians. :)

    I start talking about the special radians by using Math Teacher Mambo's em(pi)nadas analogy, except I take it even further (and I call them quesadillas due to the popularity of Moe's in our town):
    in the 2nd quadrant, I eat almost a whole quesadilla except one slice (numerator is one less than denominator)
    in the third quadrant I eat a whole quesadilla plus one slice (numerator is one more)
    in the fourth quadrant, I almost eat two whole quesadillas (numerator is one less than twice the denominator. Or, as we end up saying, it’s a weird one)
    Then we also discuss if you cut something into thirds, that's a really big piece--matches with tall triangle; if you cut something into sixths, that's a really tiny piece--matches with short triangles.

    Also, I hate to admit it took me this long to realize this (and it wasn’t until a student pointed it out)—I already knew anything ending in 5 was 45, but I did not realize that (ignoring the ending zero), anything divisible by 6 was a 60 degree, anything divisible by just 3 was a 30.

    I have my students memorize the 30-45-60 chart for sin/cos/tan--it should take them about 10-15 seconds to write down at the most. I found better success with that than the hand trick, since my students could not remember which was sin/cos and what to do for tangent, but I do show it to them in case it clicks for some students. I also suggest that they write a tiny x/y axis with the special coordinates (0, 90, 180, 270) labeled and ASTC written in the quadrants as well.

    This is the first year I've taught precal, and when we started trig again most of them did not remember specific values, but by the time we were done with 6 weeks of trig stuff, most of the students had them memorized, which made me feel a little less stressed about making sure they have them down cold in Alg II.

  14. I think too much emphasis is placed on special angles and not enough emphasis on how the unit circle is used to define sine, cosine, and tangent, and the resulting symmetry.

    If you told your students that cos(20 deg) was about .94, could they reason out values for cos(-20 deg), cos(200 deg), sin(110 deg)with a unit circle sketch (non calculator)?

    Could they rank the following from least to greatest without using a calculator: sin(50 deg) tan(50 deg) cos(50 deg) tan(1) tan(89) 0 1 5

    As far as special angles go, if they know the coordinates for the points at 30 deg & 45 deg, they can easily get the coordinates corresponding to any other special angle. I prefer using the symmetry of a circle rather than reference angles, but either will work.

    That is my 2 cents.

  15. I do pretty much what you've described, big on the board.

    It's also fun to later take that same big unit circle and walk them through what the f(x)=sin(x) and cos(x) functions will look like as you go around. (Like THIS, I mean.)

    Seems to bring gains later when we shift to sinusoidal applications.