Two years ago I developed this question that I think really helps students see how the circle comes together. It originally stemmed from students memorizing THE Unit Circle with it's 16 (or 17 if you count 0 and 2pi differently) special angles and coordinate pairs (although they often mixed up the coordinates since the "patterns" weren't quite what they thought). Later they would begin to believe that the circle only held those 16 angles and other angles just popped out of the calculator.

So, the question goes like this:

- Draw a right triangle in Geogebra. Use it to measure the lengths of all three sides and the angles.
- Using the ideas from similar triangles, shrink or expand your triangle so that it will fit within the unit circle. [This means to divide each side by the hypotenuse, but we'll have gone over the idea of similar triangles and ratios before a quiz.]
- This triangle can be flipped and flopped to fit into the unit circle in 8 different ways. Draw a picture with all 8 triangle orientations. [It looks like a double-winged butterfly with all right angles against the x-axis and hypotenuses radiating (pun intended) from the origin.]
- Find all associated 8 angles and their coordinate pairs based on your triangle.

For example:

- 40º (0.766, 0.643)
- 50º (0.643, 0.766)
- 130º (-0.643, 0.766) [Got this from 180º - 50º]
- 140º (-0.766, 0.643)
- 220º (-0.766, -0.643) [180º + 40º]
- 230º (-0.643, -0.766)
- 310º (0.643, -0.766) [360º - 50º]
- 320º (0.766, -0.643)

I think this question really helps students see the utility of the unit circle. And how we could use it to figure out any of the 6 trig functions for any angle if we were precise enough with a compass, protractor, and a ruler.