Venturing into the Unit Circle and Graphs

One of the super fun things about teaching a new course is that I get to think through what you teach from scratch. It’s also one of the hardest and most frustrating things. Because it’s new, I have nothing to fall back on! But because it’s new, I have fresh new vistas open. The world is my oyster.

Recently I’ve been introducing trigonometry to my precalculus students. They had been exposed to right triangle trigonometry, but other than a two days beyond that last year in a rushed unit that they don’t remember, nothing else.

So for a short while I’ve been creating new materials to teach trigonometry. I’m going to post them below and explain the intention of each of them. For almost all of them, I had students work in groups, and I would circulate. We would come together and talk as a class, but they were doing most of the heavy lifting. My favorites are Sheet 1, Sheet 6, and Sheet 7… so if you’re just looking for a few good things… I have embedded .pdfs of each of the sheets, but if you want to download the .doc files so you can edit, they are waaaay at the bottom.

My Trigonometry Unit (so far)

Sheet 1: This sheet is the introduction to trigononometry. I didn’t call it that. I wanted students to start seeing circles, angles, and draw the connections themselves. As they did this, they naturally had to build triangles and review their right triangle trigonometry (SOH-CAH-TOA) to answer the questions. They also were forced to start thinking of angles in relation to circles, instead of just angles in relation to triangles. As we progressed through the unit, I very explicitly started talking about moving away from the triangle to understand angles and towards the circle…. But for this sheet, I didn’t say anything of the sort.

Sheet 2: This is a sheet that is our first official foray into trigonometry. It is intended to remind students their basic right triangle trigonometry, and how to use that to find both sides and angles. It also has students derive the formula for arc length and the area of a sector. If I wasn’t teaching the advanced class, I don’t think I could have expected students to have derived the formula on their own — but for my kids, this was easy. We haven’t yet talked about relating trigonometry to the unit circle yet. But we are looking at angles in relation to circles. (Sorry about the scanned images… I hand drew stuff in this.)

Between the previous work and the next work, I gave an impassioned diatribe called “WHY DEGREES?” where I argued with brio that radians are the most natural and beautiful way to measure angles. I think I got students to at least recognize that there is something elegant in them. At the same time, I got to introduce the unit circle — because the unit circle is the thing we use to define radians! At this point, I start emphasizing the use of the unit circle to all our angle work.

Click to see the rest (there are a lot of Scribd documents embedded)…

Sheet 3:  This is a sheet that builds off of our derivation of the arc length formula … by using that to understand linear and angular velocity. The other teacher and I both agreed we didn’t want to focus a lot of our attention on this, so we just skimmed the surface of this.

At this point, we constructed unit circle plates to practice radians and degrees. Arts and Crafts! At this point, I still haven’t mentioned how to find \sin(330^o) for example. 

Finally, after they have constructed these plates, we generalize sine, cosine, and tangent to quadrants other than one (cosine corresponds to the x-coordinate on the unit circle; sine corresponds to the y-coordinate on the unit circle; tangent corresponds to the ratio of y/x on the unit circle). I was nervous that they would balk at the new non-SOH-CAH-TOA definitions I was giving them for sine, cosine, and tangent, but they totally bought it. I had them practice finding sine/cosine/tangent using their unit circle plates. We talked about the ASTC mnemonic to remember the signs. And after  they practiced a bunch (with and without the unit circle plates), I showed them the “hand trick” that Kate Nowak showed me. Honestly, there were kids in there who were hyperventillating because they were so excited about the “hand trick.” (And since we’re being honest, after learning the hand trick, I’ve pretty much used it exclusively.)

Sheet 4: This was a sheet I gave my kids to practice. It also had them practice using their calculator and converting between radian and degree mode. There is nothing really special about this sheet.

Sheet 5: I wanted to at least expose my kids to cosecant, secant, and cotangent, so I just gave them the definitions and had them practice. There is nothing really special about this sheet.

After this point, I had my students take an assessment on what we’d learned so far. Unfortunately, it was sometime in the middle of this material that Hurricane Sandy interrupted our classes for a week… so even though it wasn’t totally disjointed, it didn’t have the continuity that I hoped. After this assessment, we had Thanksgiving Break. 

Sheet 6: This is the latest packet I’ve done with my kids. Of all the things I’ve done this year, this is one of the most successful. It has students slowly start to solve basic trig equations like \sin(\theta)=\sqrt{2}/2 and \cos(\theta)=-1.2… first it has them solve them with their unit circle plates. Then it is scaffolded to help students solve these equations without the unit plate… And finally, it gets them to recognize that the “unit circle” goes on forever — so that you can have coterminal angles. Watching kids work their way through this sheet along with the discussions they were having… wow! I loved. We didn’t yet get to the graphing sine and cosine but we’re going to do that tomorrow. Hopefully it’ll go fast with the help of the graph paper that @ultrarawr specially created for me (because he’s awesome!).

Sheet 7: I haven’t tried this packet out, but after the success of my last packet, I wanted to capitalize on it. The goal of this sheet is to help students continue to be hammered with the fact that the unit circle and the graphs of sine and cosine are really one of the same. And it continues using basic trig equations to do this (but without special angles). Finally it has students look and think a bit about the graphs of sine and cosine, and learn to use their calculators to actually practice graphing trigonometry.

That’s all I got for now. More to come with graphing trig functions and transformations, I’m sure.

PS. My sheets in .doc form so you can edit are: Sheet 1Sheet 2Sheet 3Sheet 4Sheet 5Sheet 6Sheet 7



  1. My shortest and simplest advice is “say cosine first”. By saying “cosine and sine” instead of “sine and cosine” it helps kids remember that cosine is the x-coordinate and sine is the y-coordinate. In particular, use cos^2x + sin^2x = 1 and tie it to the circle equation x^2 +y^2 = 1 and you’re way ahead of the game.

    For solving stuff like cos A = 1/2, if kids remember that cos is the x-coordinate, it’s like solving x = 1/2 on the unit circle. Have kids DRAW a vertical line on top of the unit circle; then they’ll know exactly what quadrants the solutions are in.

    Same for Y = -1/2 matching sine. And tangent is Y/X… the slope… so solving tan A = 2 means drawing y = 2x on top of the unit circle and finding the intersections.

    (All advice ripped off from CME Precalculus Chapter 1, trig with radians.)

    1. Saying cosine first is probably a good idea. I admit I tend to start with sine, out of the prejudice that the co- prefix sounds like it ought to be the second in the set, but pairing the x and the cosine and the y with the sine is certainly more sensible.

      1. It’s also a huge positive for making charts of the functions. With cosine first, the columns of the chart are the X and Y coordinates, in the correct order, then the slope of the line from (0,0) to that point.

  2. I’m in the same boat as you. Teaching alg2/trig for the first time and writing to common core, updating the curriculum, etc.

    i did have an idea to introduce the graph of a trig function. i want to do the unit circle on a foam circle and graph out the curve by rolling the foam circle (with a marker pushed through it)

    S. Hills

    1. Don’t understand, Scott. Would the circle be on its edge, or flat? If you only move the circle one way (horizonally OR rotating) you either get a constant function or a circle. How would you do both simultaneously with this foam circle?

      Here’s a video I made for my students using a Geogebra animation of unit circle/trig function if you’re interested.

  3. aww :( is there a way you could email a picture of them to me? I am a first year teacher and I like the last two worksheets you have, 6 and 7. I am using those and it seems to be working well :) thank you!

  4. These worksheets are great, and I have used them as is in my trig classes with pretty high success. I am curious if you have done any revisions since using them initially?

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