Now, with all of that being said, I've never actually taught a pre-calculus course where the majority of trig. is taught. After I graduated I had a long-term sub position where I taught an advanced pre-calc, but as I look back I can't really consider what I did teaching. I hereby issue a formal apology to the students I had back then: "I apologize for not showing all of the awesome things you can do and explore with trig. I failed you as a teacher and I hope that you had someone else in your future that was able to clean up the mess that provided you."
Next year I have the pleasure of teaching pre-calc (I'll actually be team teaching it, which is even better) and I am extremely excited about it. I've been talking to my colleague that I will be teaching alongside about the curriculum and we've developed some pretty exciting stuff. Our goal is to get our academic students to demonstrate some high quality thinking and problem solving. Everything we're going to do next year with trig. will be based off of the unit circle. The plan is to spend enough time developing the unit circle and then use it to create the graphs of the trig functions and go from there (obviously this is a real brief description of what we'll be doing throughout the course). The diagram below is a goal of ours. If we can get our students to develop this diagram on their own and fully understand what it represents, we're golden.
This diagram pulls everything together in mind. It shows the definitions of sine and cosine, it illustrates the meaning of sine ('half chord') and cosine ('complement of sine'), it relates the tangent function to a tangent line (which students often think are two different things with the same name), it gives a visual representation of secant, cosecant, and cotangent, and each triangle gives the three Pythagorean Identities. Wow! And (hopefully) if we can set the correct groundwork with our students we'll be able to get them to generate this and know exactly what's happening. There will be no memorizing formulas or values, just pure understanding.
I could go on and on, but I will save that for when I'm done actually teaching lessons.
Wish us luck!