Knocking off the rust with a little geometry

We start back to classes tomorrow. My precalculus course stopped mid-trig identity before the winter break.... We had talked about the reciprocal functions and the inverse cosine and only hinted at more identities.

I am determined to not teach as I was taught and to provide many different ways to see connections. I'm pretty sure that students learn better when I'm not in the way (How can you say that in an interview with out sounding like a lousy teacher?). When they can discover something on their own they remember it, they believe it and maybe they begin to get what I missed in math class. The beauty of patterns and logic!

I had been searching for a way to both review a little and to make all the trig functions real not just some arbitrary function I might write on the board. So I went back to geometry, geometry I never saw in class, so its new and fun for me. I'm not sure my teachers ever saw it either, but its possible that I was face down drooling on my desk in boredom during that part of class...

I had seen the picture below a few times but never stopped to really look at it.

Taken from Wikipedia

I had been thinking about this picture for a while. It seemed to me like a perfect review while learning something new. There might even be a few light bulbs going off "hey my, teacher didn't totally make up those functions!"

I created a quick "worksheet," how I hate that word, that guides them through some review of basic right angle trig, the unit circle and finally gets them to the conclusion that the reciprocal functions have a geometric reality! The intent (and hope) is that I can just get out of the way and let them learn in small collaborative groups. The magic seems to happen when they don't need me, just some questions to guide them...

The worksheet can be found as a google doc. (Still needs a little polish, but writing this was good procrastination.)