# Getting familiar with the Unit Circle

In our standard precalculus class, we’ve spent 4 days “getting ready” for trigonometry. Which sounds crazy, until you see what awesome thing we’ve done. But I’ll blog about that later. Right now I want to share what I created to help kids start learning the unit circle.

Here were the hurdles:

1. We are introducing radians for the first time this year. So they’re super unfamiliar.
2. The unit circle feels overwhelming.
3. Although I am familiar with the special angles in degrees and radians, kids aren’t. So I know when I hear 210 degrees that’s “special” but kids don’t know that yet.

Here is what we have done:

1. Filled in a “blank unit circle” using knowledge of 30-60-90 degree triangles, 45-45-90 degree triangles, and reflections of 1st quadrant points to get the points in the other quadrants.

In this post, I’m going to do here is to share what I’m going to be doing to help kids learn the unit circle.

Phase I: Get confident with angles I am going to talk about these like pizza. And to start, focus on radians.

I’m going to remind kids that $\pi$ radians is a half rotation about the circle. Then we can see that each pizza pie slice is $\frac{\pi}{2}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{6}$ radians. [The “top” half of the pizza is divided into two, four, three, or six pieces! And the top half is $\pi$ radians!]

Then I’m going to work on the “easy-ish” angles by pointing at various places on the unit circle and have kids figure out the angle. I am going to have kids not only state the angle in radians, but also explain how they found it. For each angle, I will ask for a few different ways one could determine the angle measure. Then I’m going to repeat the same thing with the “easy-ish” angles, except I am going to do it in degrees.

And then… you guessed it… I’m going to do the same exercise but with the “harder-ish” angles. Start with radians. Then again with degrees. Always justifying/explaining their thinking.

Finally, I am going to let them practice for 5-8 minutes using this Geogebra applet I made. The goal here? To focus on getting kids familiar with the important special angles. Not only what the values are of these angles, but also to get them to start finding good ways to “see” where these angles are. Phase II: Start Visualizing Side Lengths — utilizing short/long

Next comes getting kids to quickly figure out the coordinates of these special angles. We’ve already been working on special right triangles, so I think this should be fine. And then… Kids are asked to visualize the side lengths/coordinates based on the drawing. So, for example, for the first problem, kids will see that the angle is $\frac{4}{3}\pi$. They hopefully would have mastered that from the previous exercise. They also will see that if they would draw the reference triangle, the x-leg is shorter than the y-leg, so they know the x-coordinate must be $\frac{1}{2}$ (but negative), and the y-coordinate must be $\frac{\sqrt{3}}{2}$ (but negative).

After practicing with this for these four problems, kids are going to practice some more using this second geogebra applet I created. Phase III: Putting It All Together

It’s now time to take the training wheels off. No longer do I give the picture to help visualize things. Now, I give the angle. This is more like what kids are going to be seeing. They need to know $\sin(315^o)$ and $\cos(3\pi/4)$. No one is going to be giving them nice pictures!

So this is what they’re tasked with:  I have a strong feeling that breaking down the unit circle in this way is going to make all the difference in the world. Fingers crossed!

If you want the file I created for my kids, here you go (.docx2017-02-xx Basic Trigonometry #2.docx:  , PDF: 2017-02-08-basic-trigonometry-2)!

1. Joseph Nebus says:

Good luck.

I know I learned these key angles and their cosines and sines with the help of writing out tables of the trig function values, over and over. It’s a mystery to me why that worked, but I suspect I have the kind of mind that latches on to stuff that can be put into neat, kind-of-symmetrical tables.

2. nkpithwa says:

thanks for sharing the files !!!

3. Rohan Prakash says:

Thanks for sharing! I’m excited to introduce the unit circle with a version of this when I teach it in May!

4. lsquared76 says:

You’ve come through for me again! Thanks Sam!