Angles and the Unit Circle can be a tough one. Not because mastering the Special Right Triangles is difficult but because you are relying on their Geometry Teacher to have done an excellent job of establishing that mastery.
How to Master The Unit Circle
With that said the only way to master the Unit Circle is to know the Special Right Triangles inside and out! However, there are MANY little “tricks” (and I use that term loosely because they are not the answer to mastery) you can use to help your students trigger recall. One of these little fun activities is the Unit Circle Hand Trick and we wrote about how to do that activity on our Geometry Coach site. The article can be found here: http://geometrycoach.com/the-unit-circle-hand-trick/
Do Not Let Your Students Memorize The Unit Circle
Aside from these types of activities there is only one way to get your students to mastery of the Unit Circle. Pound it into their head that they are NOT to memorize the Unit Circle! Not only is this a waste of time but it also doesn’t help them at all with when and how to use it for lessons later on. Think of it like this, having them memorize the Unit Circle would be like their Geometry Teacher having them memorize the Special Right Triangles instead of knowing how to create them and use them. When they do that it makes your lesson on the Unit Circle awful because you then have to go back and teach them mastery of the Special Right Triangles (we also have an article on Special Right Triangles that can be found here: http://geometrycoach.com/special-right-triangles-the-foundation-of-everything-trig/
Learn to Create The Unit Circle
They must learn to create the Unit Circle based on the Special Right Triangles. They should be able to draw it without using patterns or any other form of memorization. It would be worth your time as practice (once they have learned the material to have them recreate the Unit Circle from nothing) to have them draw the Unit Circle on paper while you walk around the room monitoring them. You will be able to tell who is following a memorized pattern and who actually knows how to create it.