I’m sure someone else had this idea long before me, but I didn’t consciously take it from anyone in particular. This year, I wanted to help my Pre-calculus students understand the patterns inherent in the unit circle. First I used this idea from Kate and Riley.

Since this activity didn’t make it clear how to get the radical forms of the key values for sin and cos, I wanted another activity to make those patterns evident. We spent a day reviewing special right triangles, and then the next day, I gave them this blank copy of the unit circle, having them fill in the degree measures first. Then I gave them quarter sheets of orange copy paper.

I gave these directions:

- Line one edge of the orange paper up to sit on top of the x axis.
- Align the right edge of the paper with the point is formed by the radius in in the 30 degree angle.
- Trace the radius (through the paper) in order to make your triangle.
- Turn the orange paper 180 degrees to get another corner (to save paper!)
- Repeat steps 1-3, but using the radius from the 45 degree angle.
- Cut out both triangles.

After that, we labeled the angles, and they were able to tell me that the hypotenuse was 1 unit because of earlier discussions on unit circle. We derived together the lengths of the missing sides through the formulas related to special right triangles to label the sides of the triangles. Have them label both sides of the orange triangles, so they can flip them and still read the measurements in other quadrants.

To get them started, we went to quadrant one and placed the 30-60-90 triangle in place. This helps them visualize the x-coordinate and y-coordinate of 30 degrees. I took them through the first quadrant and up to 120 degrees. In each class, someone was able to predict the coordinate sign changes that the x-coordinates are negative. At that point I left them to the task of completing the unit circle. I also gave them this handout to help them process through the patterns that they found. Here are some pics for the visual folks out there.