Introducing Inverse Trig Functions

So I've been reading a bit about inverse trig functions on line and how people approach teaching them. It got me thinking about what my focus is with inverse trig functions and inverse functions in general in PreCalculus. My thinking is that students in Algebra 2 learn about inverse functions and the mechanics of inverse functions (how to create graphs, how to create inverse function equations, what they mean in terms of real life situations). PreCalculus is a time to dissect the idea of inverse functions more deeply in a bit of a more theoretical way. Are there limitations to the inverses of functions? Is every inverse of a function a function itself? We explored these questions back in the fall and it was hard for students. I have a good set of classes, honors level but they have been used to learning processes and procedures and not really developing a deeper understanding.

I like to start out these lessons on inverse trig functions by having a discussion about the difference between inverse operations and inverse functions.  This sort of maps out our discussion (I lead the discussion with questions - what is an operation? what is the inverse of that operation? Let's consider an operation we all know pretty well - "squaring" - what does that look like as an operation? as a function? as an inverse operation? as an inverse operation?).

Then I steer the discussion to trig ratios as operations, as functions, etc. I try to do this by asking as many questions as possible.

Then after the discussion I distribute an "explore" packet to students. We do the first page together (it's just a review of inverses with quadratics and the domain & range restrictions). Then students work together in their small groups and using their graphing calculators work on visually representing  and defining domains and ranges of inverse trig functions.  I put a big focus on the visual (it's how I learn best and I find students often learn better that way too),

I thought I'd visually go through what we do & students do with the explore packet. You can open a blank electronic version of this worksheet at the end of the post.

We use TI-84 or TI-83 graphing calculators. We graph a basic quadratic and state the domain & range:

Then using the DRAW menu we can actually see the true inverse without restrictions of this function. We do that and draw it. Discussion ensues with "what's wrong with this picture?"

From there we remember that the inverse of the quadratic is the square root function. We draw that. We carefully examine the graph and the function equation to be clear about the domain and range. 

A little "mantra' we chant is the the domain of the inverse comes from the range of the original. This is verified and we see how it restricts the range. All this is review of something we did back in the fall.

Now students get to work on their own doing a similar process to sine, cosine and tangent. These bits below show ideally what they should be coming up with. If there is time I have three small groups present what they come up with for each and explain. Then I lead the discussion into how these relate to the unit circle. That will continue in my next post (we pull it all together in some ISN inserts).

SINE:

COSINE:

TANGENT:

Is this good math teaching? I hope so. I just remember when I first started teaching PreCalculus 10+ years ago. I was equipped with my previous learning as a precalculus student many many years previous and a bunch of textbooks. No real curriculum. When I got to inverse trig functions I had to ask myself - what are these really? What do I want to know about them? How can I make sense of them in relationship to everything else I know about mathematics? I have a hard time using problem sets in textbooks to drive my teaching. I like having a deeper understanding of how things work rather than just know how to solve given problems.

We do then go onto apply these ideas to problems. But I'd like to think of another way to assess understanding than just evaluate inverse trig functions. Maybe some writing prompt for the students to respond to.

All my files on Inverse Trig Functions can be found HERE (and this is everything - introduction sheets, evaluate sheets, ISN materials)