Friday, April 6, 2012

VFC: Intro to Trigonometry

Here is a Virtual Filing Cabinet for an introduction to unit circle trigonometry. The goal of this unit is roughly to get students conversant in the sine, cosine and tangent functions. Following this unit in my class is a full unit on graphing the trigonometric functions.

What am I missing here? Point me to your favorite trigonometry resources in the comments.

[Last Updated: 4/5/2012]

The Hard Parts

Part of what makes this unit challenging for me is that students may or may not come in knowing the right triangle trigonometry definitions of sine and cosine. Some teachers like to start with the right triangle definitions and expand them to all real numbers. Because I know that I can't rely on my kids remembering the old definitions, my solution is typically to offer a fresh start with the circular definitions of the trig functions, and then later tie them back in to right triangle definitions.

What I like about trigonometry is that so much of it can be connected to one central model -- the unit circle. This is empowering. But kids had better know their unit circle stuff, and so kids really need a firm grasp of the unit circle by the end of the unit.


This song goes first. It's fun:

Kate and Riley both have good starting points for the beginning of this unit, but my understanding is that they start with the right triangle definitions of sin, cos, and tan. I don't like to start there for two reasons. First, because my students usually don't remember those definitions. Second, because I'd rather give them a clean foundation and connect it to the right triangle definitions later.

I detailed my preferred starting point in a post:.

In general, I like the worksheets and implied approach of the eMathematics textbook. I use their worksheets on angles and rotation terminology after I've reviewed the equations of circles with my classes. You can find them at the link below, though everything from their Lesson #6 and on I consider part of my next unit, on graphing the trig functions.

I don't know where I first saw that the tangent function is the slope of a radius in the unit circle. I never learned it that way, but it makes so much sense to me and my kids than just defining tan directly in terms of sin and cos.

Here's a full course through problem sets. I haven't looked through it carefully yet, but it looks good:

Oh yeah, radians. And if you're in NY, arc length. I spend a day converting unit with my kids, and radians <---> degrees is an application of converting units. Sometimes I use this not-so-great resource that I made that just contains some goofy units of measurement to shake kids out of their comfort zone.

I'm torn as far as converting units is concerned. On the one hand, there's a procedure for converting units which can help you convert when you don't have an intuition for the relative sizes of the units. At the same time, gaining intuitions about the relative sizes of units is helpful. So my approach with radians/degrees is to both things. First I try to move them out of their zone of intuitions with an activity, and then they calculate "about how many degrees are in a radian?" to give them intuitions. I've also found that kids in my classes get confused about radians. They say things like "The unit circle has 2 radians" as opposed to "2 pi radians." What I've learned to do is ask them to estimate the number of radians as around 6, and ask them to make back-of-the-envelope calculation with that number, and then use pi for the more precise calculations.

Also, I teach arch length as a proportion problem. Meaning, what percentage of the circle is 2 radians? If the radius is 5, what will the circumference of the full circle be? What percentage of the circumference are we looking for? I know that's probably a no-brainer for a lot of you, but I had trouble coming to that approach.


  1. I like your thinking, and really appreciate all of the useful information you post, so I hope you don’t take these comments the wrong way. A lot of it probably address some of the unease you expressed with the amount of scaffolding to provide.

    I would like students to be proficient with the right triangles definitions before beginning the unit circle unit (warm-ups or homework a couple of weeks before unit circle). But, I agree that it is better to use the unit circle definitions first in the unit circle unit.

    Why not deal with radians and arclength before your right triangle intro? Also, why do you need to review special right triangles after day 1. Simply label the coordinates corresponding to 30 deg & 45 deg on the unit circle and say that for whatever reason a lot of people make a big deal about these angles. There is no more reason to review special right triangles now than during an algebra 2 unit – the development has nothing to do with the development of the unit circle.

    Day 4: Why the tedious table in question 1? Now you have another tedious table, the same table, in question 4. Also, it seems too early to introduce a graph. Why not address the symmetry of the unit circle. If you know the coordinates corresponding to one angle, for what other angles can you find coordinates.

    Day 5: You are just handing over the Pythagorean identity on a platter with question 1 (and it is tedious to boot). Skip question 1 and instead of question 5 & 6, ask if they can find a relationship involving sine & cosine – you may well need to give hints. Question 2 is silly, ask things like is there an angle where the sine value is not defined, is there an angle with a minimum cosine value, which angles have relatively big sine values, how does the cosine value change as the sine value changes, do any angles have negative sine and positive cosine values, etc. Why insist on simplest radical form?

    1. All criticism is taken well here. I'm so grateful for criticism, you have no idea...

      Anyway, I don't get how you would have students find the values of the x and y coordinates of points at a 30 or 45 degree angle without special right triangles?

      As far as the tedious table, what I like about that is that students quickly are motivated to start looking for shortcuts, and then they notice various symmetries. Most of my students find symmetries through that tedious question, opposed to explicit questioning.

      As far as your last point re: the pythagorean identity, I'm not sure that I see my students just proving the pythagorean identity out of the air like that. I felt -- and still do feel -- that students need a bit more support for the Pythagorean Identity than you'd give them. Though, I really like your question about how sine changes as cosine changes.

      I feel like I'm not getting how your lesson on the pythagorean identity would look. Do you have something written up, maybe?

      Also, the questions with simplest radical form are intended to match up with questions from NY's Regents exam that require using the Pythagorean Identity in that way.

      Thanks for the incisive comments.

  2. I see your point about students looking for short cuts to complete the table, but I still don’t like too much focus on special angles. The unit circle really isn’t about special angles, it is about the definition for sine, cosine, and tangent.

    Last year, gasp, I just told them the coordinates for the special angles. There is nothing special about these angles as far as the unit circle goes – and the special triangle proofs won’t give students any insight on the unit circle. I am not proving the Pythagorean Th., so I am not too worried about proving the special triangle relationships.

    OK, I am re-thinking one comment. If you want to introducing the Pyth. Id so early, you will probably have to give a lot of support. I think it is better to introduce it later with almost no support. I am not sure what method you use to solve the typical unit/non-unit circle problems (given a cosine value find the sine value, etc). I use a diagram of a unit or non-unit circle with a right triangle for all of this – we constantly go back and forth between the diagram and the sine/cosine/tan value. I don’t need any Identities until I start the verification unit (next chapter in most books & not my favorite). So, when I am ready to develop the Pythagorean Identity in that next chapter, we have used the idea so many times that they don’t need support.

    One way to go is briefly describe and give an example of an identity: 3(x + 1) = 3x + 3. Then ask for identities involving sine, cosine, and tan. You will get a lot of them. Be ready to prod them to be more sophisticated. Give a hint if you need to for the Pyth Id.: say “Pythagorean Theorem”, for example. Or, just tell them you have one more you are not sure about, the Pyth. Id, and ask them to prove it to be true or false. I would not use a handout for this, collect ideas as a class, put them to work, regroup to share ideas, etc.

    This is the sort of thing I use during the first two or three days: