# Inverse Trig Functions

At TMC13, I was in a group of people talking about precalculus. One of the exercises we did was make a list of some of the topics we found challenging to teach as teachers — and we broke out in groups to try to come up with ways to tackle those topics.

My group’s topic was inverse trig functions. (This was with April, Dan, Greg, and Andrew.)

Our initial task was to find the deep mathematical idea behind the topic… why we teach it, what we think we can get out of it conceptually… and what we sort-of converged on is that the topic really illuminates the idea of inverses and restricted domains. And that’s about it. And when push came to shove, we decided we didn’t find that restricted domain is something we really care about. We decided we didn’t really care about the inverse trig graphs, and the work we put into that side of things wasn’t really worth what we little we were able to squeeze out of it. It’s not that it is horrible, but we just didn’t couldn’t justify it.

So, honestly, we decided to just focus on inverses, and the idea of them as “backwards problems.”

Thus, we came up with two things:

1. A packet that has students secretly engage with inverse problem work before they even know what they’re doing. So the first packet is meant to be used before any unit circle trig is introduced. (A few of us, especially April, did something similar in her classes, and randomly, Greg Taylor did a my favorites on the same essential idea!)

In fact, if I were to use this in the classroom, I would not even mention the words “trigonometry.” I would focus on the idea of coordinate planes and circles, and simply leave it there.

2. A packet that students work on after they learn unit circle trig — and that more formally introduced the idea of the inverse trig functions. It tries to draw connections between the unit circle, the sine/cosine graphs, and their calculators.

There are concept-y questions for both packets. I’m including both packets below in one document. I’m posting one with a few teacher notes, and one with the teacher notes hidden. (The .docx is here if you want to edit!)

Packet with teacher notes

Packet without teacher notes

We did all this planning in pretty much an hour and a bit — from start to finish. And then I pulled together the ideas to make this document. I’m not sure I was able to capture everything we talked about, but I think I got most of the big things. Apologies to my collaborators if I totally botched the translation of our vision to reality!

1. Niki says:

Love this! Thanks for sharing. I may use this with my Algebra 2 classes this year. What did the other groups come up with?

2. Michael Paul Goldenberg says:

Of course, this makes me wonder about motivating the topic in a more general way: why are inverse functions important (if they are)? Is it worth starting (or finishing) with some conversation about that broader question?

3. Probably! Do you have any thoughts on how to frame it in that more general way?

I see this much like getting the need for the log function… To do that, I give a sheet that Maria Andersen has, which is simple but powerful.

It’s like (just making this up quickly now to give the idea):

EQN: SOLVE : inverse operation used to solve
x+3=5 _________ ____________
x-6=7 _________ ____________
5x=3 _________ ____________
x/2=8 _________ ____________
x^2=4 _________ ____________
2^x=7 _________ ____________

And then when they get to the last one, they realize they can’t solve it algebraically… because they don’t have an inverse operation…

Sam

4. I have long thought that the whole inverse idea is the ‘dirty little secret’ of Algebra I and Algebra II and that we do not give it sufficient time in the daylight. I’ll joke with my kids that every time I teach them anything in these courses, they can expect to be taught how to undo what we’ve just done. Adding/subtracting, multiplying/dividing, distribution/factoring, exponents/radicals, variable exponents/logarithms.
The graphs of the inverse trig functions bother me even more than it sounds like they bothered you guys. I have seen texts with different conclusions about the graphs of inverse secant or inverse cosecant. I try to avoid them almost completely other than a quick peek at GeoGebra (or maybe Desmos if I can fall in love with that sufficiently)

1. There’s something really neat about inverses being a core idea of high school math. I don’t normally see that listed when ppl think about what’s truly important — it is usually just something people do abstractly and then forget until it reappears (when I say “people”, I mean “me”).

But I wonder…if I were to think of it as a core idea in high school math, I would feel more of a push to talk about restricted domains when talking about inverses — because talking about the inverse of squaring would necessitate talking about restricted domains…

You’re totally right about the lack of consistency when it comes to various textbooks and the inverse trig. Yeesh.

1. Sam

Yeah, I’d have to say that I disagreed with the original post when you dismissed domain restrictions. I think that this sort of backward thinking required when engaging with inverse functions is much more rich when we consider domains/ranges and where the answers for these functions are even meaningful. Trying to squeeze some context in always helps as far as I can tell.
Your example of the inverse of squaring is spot on – that is where we can first meaningfully discuss restrictions. When I introduce the idea of inverse trig functions I play a mean game where I say ‘I’m thinking of an angle whose cosine is 1/2. What is the angle?’ And, of course, I can always tell the student that they are wrong no matter what answer they give me. This leads to a nice conversation about domain restrictions and the power of functions being creatures where there is one answer to one question.

2. I keep flipping back and forth in my opinion on them. Not whether they are a useful thing to teach per se — because I’ve realized I’m Mr. Conceptual Teacher and there are some deep conceptual things from this domain restriction stuffs — but given all that we have to cover, if something needs to go, it feels like given a choice between for example:

1) doing fun explorations with matrices and their applications
or
2) talking about domain restrictions in a deep way (with the graphs)

I would choose (1). That’s all.

5. P.S. My viewer really messed up the formatting! Try downloading, and, if that doesn’t work, I’ll send PDF’s.

6. Kevin Hall says:

Sam, while stealing things off the TMC13 site, I noticed that you had collected Precalc resources on SugarSync. Are those only available to TMC participants? I would love to copy them and see how I can use them. Thanks for this great lesson on inverse trig functions.

7. misscalcul8 says:

Do you have an answer key for this? I’m teaching this for the first time and have no idea what I’m doing. As per usual. =)