Exploration 10 
First, let's setup our function graph in Desmos (to view yourself click on the image below or click here: https://www.desmos.com/calculator/h7iub7bvb7):
We have just generated the unit circle! Have students first write an equation for a circle with a unit radius and center at (0,0):
You can also use some other depictions to explain the unit circle:
Cosine function: https://www.desmos.com/calculator/s8jg20tfws
Sine function: https://www.desmos.com/calculator/kxfekf0kgb
Your students should also be able to understand the following trig identity property from this depiction:
Cosine function: https://www.desmos.com/calculator/s8jg20tfws
Sine function: https://www.desmos.com/calculator/kxfekf0kgb
Your students should also be able to understand the following trig identity property from this depiction:
Now, let's explore some more complicated depictions:
Explore yourself here: https://www.desmos.com/calculator/8pdqk1ewkx First, let's talk about setting b = 1 and playing with a. As a increases, the period of the function seems to decrease in size and these increased waves in the function seem to wrap themselves around the yaxis. The entire structure, which almost appears like a bracelet seems to orbit around the xaxis as well. Next, let's talk about setting a = 1 and playing with a. As b increases, the period of the function seems to decrease in size and these increased waves in the function seem to wrap themselves around the xaxis. The entire structure, which almost appears like a bracelet seems to orbit around the yaxis as well.
Now, it get's interesting when we animate both a and b. When a>b, it appears that the entire structure orbits around the yaxis. When a<b, it appears that the entire structure orbits about the xaxis. The wrapping of the period changes frequently in a crossing pattern, sometimes about y=x or y=x and sometimes just about the origin. At times the same wrap and orbit patterns seem to happen about the y = x and y = x axis. At other times, the orbital or wrap patterns seem to almost happen about what might be the z axis if we were working in R2. It's quite fun to watch though!

