Daily Archives: April 15, 2008

I love it when my students think for themselves. 

When learning the law of sines and law of cosines — and when it’s appropriate to use one or the other — our textbook gives pretty prescriptive directions. For example, when you are given SSA (a side, a side, and an angle opposite one of those given sides), you are supposed to use the law of sines. And depending on these values, you can actually get two possible solutions!

Let’s work this out.

If in triangle ABC you’re given that side a=6, side b=8, and angle A is 40 degrees, then let’s solve the triangle.

Using the law of sines, . Rearranging, we get . But we know that sine is positive in quadrants I and II, so we get for or . And hence, we have two possible angle values for B, which leads to two different triangles. [For a more detailed explanation, see here.] Another quick application of the law of sines yields that or

And looking at the picture below (cribbed from the site above), you can see that both triangles are possible!

Here’s where student thinking is awesome. The book, as I said, says that everytime you have SSA you should use the law of sines. And I agree, it is easier. But it is definitely possible to use the law of cosines too, as one of my students pointed out to me. 

Let’s do it:

This simplifies, with some rearranging, to the quadratic: . This can be solved to get the two values for , which are or .

I love that it works, and that the student insisted that we could do it. It might be slightly more work, but not that much more, and the exploration aspect is awesome. [1]

[1] An extension for a project for next year might be: how can we use the fact that we can generate a quadratic help us in determining when we have two possible triangles, one possible triangle, or no possible triangles.

In my trigonometry classes, I decided to deviate from the textbook ordering of concepts. The other teacher, thankfully, was on board. (And next year, I want to tweak things even more.)

Our textbook presentation of trig starts out with reviewing right triangles, SOH CAH TOA, and in the same section, introduces the reciprocal trig functions (csc, sec, cot). It then goes into application problems, involving angle of elevation and depression. Finally, it throws all the hard but juicy stuff into the next section — a section that took me 4+ days to cover. It included introducing the concept of using trig for angles greater than 90 degrees (a VERY hard concept for kids to grasp), reference angles, quadrant analysis, and a variety of different types of problems that students are expected to do.

But then the book starts veering into radians (which I covered already this year, and next year, which I’ll postpone), the graphing of trig functions, and finally goes into the translation and stretching of these graphs. 

I skipped this graphing work and an entire other chapter to get to the law of sines and the law of cosines. Um, yeah, hello? Let’s think about this:

Students start by learning that trig helps with right triangles and they do application problems. Then they learn how we can extend this trig work to angles greater than 90 degrees. Me thinks that it would be natural to show them how trig can help them with all triangles at this point – including obtuse triangles. (Importantly, the law of sines also tests a student’s understanding of reference angles.)

My students seem to find doing problems with the law of sines and cosines very tedious, yes, but they also love the grounding and concreteness of it. They told me that. Which makes me think it’s better to put that sort of thing at the beginning of our exploration of trig. And leave the more abstract discussions to later, when the basics are fixed in their minds.

Rethinking the textbook makes me feel good, because it means I’m paying attention to the flow of the subject, to how I’m presenting the topics, and what students are thinking as they learn trig. And it means I’ve started fulfilling one of the goals I set before school started, one that I decried just couldn’t happen my first year.

[1] For giggles, I want to share what the section after the law of sines and cosines is in the textbook: Graphing complex numbers on a complex plane. Can we all say “jarring transition”?