Students should return to their groups from the previous lesson and pick up where they left off on the Joggins Wharf problem. I will encourage them to share what progress they made at home and then work together to solve the problem. (I will also circulate quickly to check in with the groups to find out who did the homework and who did not.)
After about 10 minutes, the class should be ready for a presentation/discussion of the solution. I will have one group present theirs at the whiteboard. It's quite possible that other groups will have slightly different equations, and it is worth discussing a few of these. The problem is somewhat open to interpretation, so variation is reasonable, but we need to discuss its effects on the final answers. (MP 4)
At this point, the class should be familiar with radian measure for angles and with the use of the unit circle to extend the domain of the sine and cosine functions. They should also be familiar with the periodic shape of the sine and cosine curves. But what about tangent? [For a good overview, you might check out this webpage.]
I'll ask the class to remind me what the tangent of an angle is as a ratio. From previous classes, they know the tangent as "opposite over adjacent", which is a good starting point. I'll jot this down on the board along with the corresponding definitions for sine and cosine. Next, I'll remind the class how we used the unit circle to revise these definitions so that we could do so much more with sine and cosine. Drawing the unit circle with a reference triangle in the first quadrant, I'll ask the class what the word "tangent" makes them think of geometrically. They should reply that a tangent line is a line that just touches the circle at a single point.
Great! With this, I'll add the tangent line so that my board looks like this. Next, I'll give the students this prompt: "We know what it means for a line to be tangent to a circle, and we know that the tangent of an angle is the ratio of the opposite to the adjacent. Now, please explain to me how these two definitions are equivalent in this case." Giving the students some time (and encouragement) to think it through, I'll suggest that they sketch the picture for themselves, that they think about similar triangles, and that they discuss it with their classmates.
After a few minutes, the class should be able to walk me through a brief argument from similar triangles that tan(x)/1 = sin(x)/cos(x) = opp./adj. I'll act as the scribe for this explanation, and I'll call on a variety of students to provide just one or two steps at a time. (MP 3) By the end, everyone should see that we now have two new interpretations of the tangent of an angle: it is the length of a specific line relative to the unit circle, and it is the ratio of sin(x)/cos(x). (You can see several variations of this argument here and here and here.)
[At this point, the class is primed to recognize the Pythagorean identities involving the square of the tangent function. They'd just need to look for the right triangle! It would be an interesting aside, but it's really a subject for another lesson.]
Now that the students are comfortable thinking of the tangent of an angle as an actual tangent line, it's time to extend the domain to include angles greater than 90 degrees and less than 0 degrees.
Using an applet like GeoGebra (see the included resource) or something online (like the one here), we'll first examine the value of the tangent for angles from 0 to 90 degrees. In doing so, we'll make note of the following:
Once we've made these observations, and before allowing the students to see what the applet will do, I like to ask, "What's going to happen to my tangent line if I go beyond 90 degrees?" You see, it isn't clear how we should interpret it in the second quadrant, and it's good to entertain some possibilities before going on. I'm always curious to see what the students suggest and how they argue for or against the different options. After a brief discussion, I'll go ahead and show them what the applet does and make the argument that this length may be considered "negative" since we're measuring in the opposite direction. (Also, since tan(x) = sin(x)/cos(x), we can argue from the fact that cos(x) and sin(x) have opposite signs in the 2nd quadrant.)
Continuing around the circle, the class should notice that
There has been a lot to learn from this lesson, and it's good to end with some fairly straightforward practice. I've selected a few problems to do just this.
The first few problems ask the students to associate the tangent of an angle with a particular line & angle in the unit circle. These should be familiar and fairly routine.
The next two, however, are more complex application problems (MP 4). The first of these problems is important because it illustrates a real-world phenomena that are periodic without being sinusoidal. The second provides a geometric context in which the tangent is the most reasonable function to employ.
There will be just enough time to get started on these before class ends, and I'll make it clear that they're NOT homework (yet). I'll use this time for some quick formative assessment by observing the students as they work, asking questions, and listening carefully for evidence of understanding.