# A Beautiful Mistake

In Precalculus, we’re working on solving basic trigonometric equations. A student was working on this problem:

And he made an error on his calculator and accidentally typed $\tan^{-1}(-0.1)$. He got an output of $-5.711^o$. I think he realized his error when comparing his answer to his partner, who typed in the right expression into his calculator: $\sin^{-1}(-0.1)\approx -5.739^o$.

And his curiosity was piqued. Was it a coincidence that the two results were the same?

Of course my curiosity was piqued too. How could it not be? And his question led me to trying to figure this out on the fly. Why were the two results so close? A difference of about $0.028^o$. I tried to wrap my head around that… Even in the context of these $5^o$ results, that is so miniscule!

So in this short post I’m going to share what I did at this moment. In total, this took about 3 minutes.

1. I acknowledged it was so bizarre that the two results were so close, and that the question of why that might be was an awesome question. I said to the student: let me share how I’m going to think about this with you, and maybe we can figure this out.
2. I throw desmos on the screen. The rest of the kids are working in their groups on something else, so I’m just working with this one kid and his partner. I switch desmos to degree mode, get a good window, and type in the following:
3. I zoom in around y=-0.1.

and then I make the sine curve disappear, so we only saw the tangent curve. And then I made the tangent curve disappear, so we only saw the sine curve. I said: “if these curves weren’t different colors, would you be able to tell them apart?” (Leading question. Obvious answer prevails. No.)

4. So I said: it’s weird that around here, for small angles, the sine graph and tangent graph look the same. But that’s not true for most angles. So I’m wondering what it is about sine and tangent which make them both similar for small angles.
5. And then it strikes me. So I share my insight: “What is the meaning of tangent again, graphically?” And we review that tangent is slope, which is steepness, which is rise over run, which is y over x, which is sine over cosine.
6. So I write on the board: $\tan(x)=\frac{\sin(x)}{\cos(x)}$ And I say: let’s look at what happens for input angles close to $0^o$. And here he has the insight that for these angles, the denominator is really close to 1. So we’re left with $\tan(x)\approx\sin(x)$. [1]
7. I was elated at this. At the question, and positively giggly that I was able to figure it out using graphing and simple logic. And I remember saying that “This was the most interesting math thing I’ve thought about this whole week! Thank you!”

Why did I want to write a blogpost about this? Not because it was a good learning experience for the kid who asked it. I literally did all the thinking and shared my insights as I had them with him. (So it shows him he has a teacher who values his questions and enjoys problem solving, but it didn’t really push forward his content knowledge much.)

The reason I wanted to write it is because I immediately saw that this could be an amazing learning opportunity for students next year if I design it carefully. I could see spending a good 20 minutes of class on this question. I give groups giant whiteboards. I give them a prompt (which I will draft below). I have some hint envelopes at the ready. And I encourage the use of desmos (which would encourage some graphing work!).

Last year I had a student who accidentally typed something incorrectly in his calculator. He typed $\tan^{-1}(-0.1)$ instead of $\sin^{-1}(-0.1)$. He realized he had an error only after doing a super careful comparison of his answer with his partner. Their answers differed by a minuscule amount, a mere 0.03 degrees. Imagine that angle! How small that difference in angle is! This student was left wondering if this was just a strange coincidence or not. It turns out that it is not a strange coincidence, and there is a reason that the two outputs were super similar. Your task is to figure out why! Use Desmos! Talk to each other! Go to the whiteboards! Exploit what you know about sine and tangent! Figure out what the devil is going on!

What I love about this question is that its concrete, but also brings up so much conceptual knowledge. Kids have to understand what inputs and outputs of inverse trig functions are. Kids have to know what sine and tangent represent on a unit circle. Kids might even look at graphs! But I could see different groups getting at an explanation in two different ways… Some using a unit circle. Some using desmos like me. And maybe some using some method I haven’t thought of!

I also thought what a fun question this could be if translated for a calculus class. A consequence of the fact that the graphs look the same for small angles is that their derivatives will also look the same for small angles. And also the taylor series approximations for sine and tangent will be similar-ish — for the lowest order term, in any case!

[1] Admittedly some handwaving here. That’s why we have calculus!

1. I love this post. It reminds me of a conversation I had in my geometry class last week. I usually show kids a good old-fashioned trig table, and we spend a decent amount of class time just noticing stuff about the table. One thing that kids noticed was that sine and tangent start out in the same ballpark…and then right around 60 degrees tangent goes nuts and starts accelerating like mad, while sine is still slowly making its way up.

I also love the move you make, turning a mistake into a new bit of math. I think it shows a really nice way of thinking about mistakes. It’s not that mistakes are valuable because they’re wrong and wrong stuff is great in math. (This is a view that Jo Boaler comes very close to suggesting, I think.) It’s that mistakes are normal in math, and that in math is a sort of search for interesting things to think about.

Why shouldn’t we find interesting things in mistakes? In math, we’re desperately seeking problems. Among other reasons, mistakes should be normalized so that we are able to THINK about them, without freaking out and sweeping them under the carpet.

Thanks for the post!

2. Mr K says:

In my algebra 2 class, we cover both the unit circle and sequences and series. It is a natural thing to introduce the taylor series expansions for sin & cos, which in turn makes it easy to see what the first order (i.e. small angle) behaviors are for them.

This is taken advantage of for physics problems, which is why I’m excited to be teaching a combinations physics/calculus class in a couple of years.

1. Cool! I’m curious, do you just *tell* your Algebra II kids that there is a polynomial approximation for sine/cosine? Or do you actually delve into the ideas (sans calculus)?

3. Hey Sam!

Excellent teaching post about the snap decisions teachers have to make, and how important it is to train your gut to be attuned to these sometimes fleeting opportunities.

Just some additional information about your and your student’s discovery: This is called the small angle approximation in physics, as a rule of thumb it works for angles <π/6. The best way to see this behaviour is to do a Taylor expansion about x = 0 for each of these functions and compare the results!

1. Hi Nathan! Thanks for your comment. I remembered learning – ages ago – about the small angle approximation. Probably in high school or early college physics. But I didn’t quite remember it, so it was fun “rediscovering” it, with the same excitement associated with having a new insight! I didn’t know that people used that pi/6 as a rule of thumb. That’s good to keep in my back pocket, because I did wonder at what angles I would probably say “they weren’t close enough”!