# Multiple Representations for Trigonometric Equations

I have to say that we’re doing some pretty neat stuff for trig this year in precalculus. I’m working with two other teachers and totally writing everything we’re doing from scratch. I had about 3 days to teach solve some basic trigonometric equations. They are basic. Like $2\sin(x)+5=4.7$. But we’ve put a lot of thought into what we’re teaching, how we’re teaching it, and why we’re teaching it — and more complicated trig equations just didn’t make the cut. [1]

Besides not-a-lot-of-time, the other bugaboo I was contending was how to deal with inverse trig. Long story short, I’ve found a way to teach inverse trig which makes me fairly happy in my advanced precalculus class. But because of our time constraints, I decided that we could get my standard precalculus kids solving trig equations without understanding the theory behind the restricted domain of inverse trig functions. :) Why? They learned years ago in geometry that if they have a triangle like the one below

they could get an angle, like angle $A$, by writing: $\sin(A)=\frac{3}{5}$. And then using the inverse of sine, they could get $A=\sin^{-1}(\frac{3}{5})\approx 36.87^o$They know about the inverse trig functions already. So I wanted to exploit that fact.  And if organically a question about what the calculator was doing when spitting out an answer, and why it only gave one answer, I promised myself I would address it. (This year, no question like that arose.)

A quick last note, before I shared how I approached these few days in class, I decided to totally eliminate the use of the term “reference angle.” Kids would discover the relationships among the solutions of trig equations on their own. No need for new terminology here. Just logic.

Day 1: Three important “do nows”

This led to a great discussion. Every group decided the “top left equation” was going to be the easiest. And every group decided that the log and tangent equations were going to be the hardest. When I pressed them on why, they said it’s because they forgot logarithms from last year, and that tangent was just kinda tricky. They could “undo” a square root or a square, but they didn’t really know how to “undo” a logarithm or tangent function.

Next I threw up this slide. I just wanted to remind kids that sometimes there are more than one solution to equations — even simple equations they know. I also wanted them to see that they knew something about the tangent equation. They knew it had infinitely many solutions — even though they might (right now) know what those solutions are!

Finally, I wanted to do a serious review of special angles and their relationship with the unit circle. So I had kids spend 5 minutes solving these basic trig equations.

Obviously I put the unit circle on there as a prompt to get them thinking. And YES, that last trig equation, with the 3/7ths, was done on purpose. I asked kids after they got stuck on it if there were some of these they would not want to appear on a pop quiz. They all recognized that the 3/7th one was bad because it wasn’t one of the coordinates associated with the special angles.

This laid the groundwork for the packet.

[docx editable version: 2017-04-24 Basic Trigonometric Equations]

Kids had good conversations and were able to solve equations like $\sin(x)=0.3$ and $\cos(x)=-0.8$ using the unit circle/protractor, a detailed graph of the sine and cosine waves, and using their calculators to get fairly precise answers.

Their nightly work was simply to finish the sine and cosine questions in Part 1 (questions #1-4).

Day 2: Expanding Understanding

I started with an awesome “do now.”

I thought this was going to be a quick 4-5 minute discussion. But kids took 3-4 minutes just to really talk in their groups. And I had them share their thinking. It led to kids talking about “efficiency” and “conceptual understanding” themselves! They all pretty much though the unit circle was the best way to solve it — even with the annoyance of the protractor — because they liked the conceptual understanding it provided. They thought the calculator did the work quickly, and was more accurate, but it annoyingly only gave one of the solutions (so you had to use logic and the unit circle to figure out the second solution), and you could easily forget the meaning of what you were doing. I was so proud of what they were saying. Super awesome metacognition! All in all, this was probably 7-8 minutes.

Then I let them loose on the tangent questions in the packet (Part I #5 and 6). They initially had to solve $\tan(x)=1.1$ using a protractor. Every single group remembered tangent represented slope. Most groups reasoned that if $\tan(x)=1$, they would get $45^o$ and $225^o$ as their solutions. And since this slope was slightly greater than 1, the angles would be slightly different, just a few degrees higher. It was lovely. (And exactly what I hoped would happen, which is why I chose to use 1.1 in the equation.) But one group literally drew a line with a slope of 1.1 and measured the angles associated with that. I wasn’t surprised that a group did that, but I expected a few more to do so. (I had this group share their thinking with the rest of the class, at the end of the period.)

Then kids spent the rest of the class working on select questions in Part II (8, 9, 11) and Part III (13, 14, 15).

For nightly work, kids finished any of those problems (#8, 9, 11, 13, 14, 15) that they didn’t finish up in class.

Day 3: Polishing Things Off

I started with a question that I wanted to reinforce after the previous class:

We did a bit of review of some unrelated Algebra II ideas to help set them up for our next unit on polynomials. And then…

… to work! I had kids discuss problem 13 in their groups first (since I could see that being a place where a kid, at home, might get trapped… and I wanted them to use each other to get unstuck). And then they compared their answers to the other nightly work questions — and used a solution sheet I gave them to see if they were correct. Then I set them loose on using Desmos to do Part IV. The rest of the period was spent working on finishing up the problems that weren’t assigned in the packet (the ones they skipped).

Pretty much all groups were working together amazingly, and when I went around to check in on different groups, everyone was getting all the questions correct. The biggest problem was actually finding a good window in Desmos! If that’s the biggest problem, I’m golden.

What I loved:

Okay, so I’m going to toot my own horn here. Although the packet may “look” simple, I have to say the only way to see why it’s so awesome is to actually do it. The choice of having kids solve $\sin(x)=0.3$ and then immediately solve $\sin(x)=-0.3$ was on purpose, to generate good conversations with kids about reference angles without using that term. The choice of $\tan(x)=1.1$ was done specifically to exploit their understanding of $\tan(x)=1$. And the fact that they’re constantly looking at the same question through three different lenses (unit circle, wave, calculator) is deliciously sweet. And then — at the very end — they get to see the solution a fourth way, by using Desmos to graph these equations to find a solution? SO COOL. Because the very last thing we had done in this class was learning transformations of sine and cosine graphs! [2]

This packet, and associated “do nows” and conversations, did what I was hoping for. It highlighted multiple representations. It had kids thinking deeply about the meaning of sine, cosine, and tangent. It had kids develop a way to understand multiple solutions to trig equations by simply using logic and what they know. It had kids recognize that the more they understand trigonometry, the more ways they have to solve a trig problem. And no kid got derailed because they didn’t understand inverses deeply.

[1] I could argue a case for these type of equations, as well as a case against them. But considering our goals and what we’ve already done with trig, I think we’re making the right decision. Why? Because our goal isn’t solving algebraic equations writ large, and I could see solving something like $2\sin^2(2x-180)=5$ being useful for that. But for getting a deeper understanding of the trigonometric functions? I see less value. (Not no value, mind you, but less…)

[2] We did this in a deliciously marvelous way. I hope to blog about it!