Lesson 2 of 8
Objective: SWBAT explain radian measure as the length of an arc on the unit circle. SWBAT convert from degree to radian measures.
Extending the Domain
Based on the practice problems from the previous lesson, the class is now prepared to extend the domain of the trigonometric functions to include all real numbers. All we need to do is use the unit circle to show that it is reasonable to include angle values greater than one full rotation and to assign a meaning to negative angle measures.
I will ask the class to begin by taking a couple of minutes to check their solutions to the practice problems with one another. Together they should be able to catch some of the simpler mistakes.
After a few minutes, I'll begin calling on students to read their answers one-by-one. (On this set of problems, I'll accept decimal approximations for the radical values, but I'll make a point of asking for the exact values, as well (MP 6).) If the entire class agrees (and they're correct), then we'll move quickly to the next problem. There are a few things I'll emphasize along the way:
- How to sketch the reference triangle once we leave the first Quadrant.
- The repetition of the two special right triangles.
- The correct interpretation of an angle greater than 360 degrees. (Can anyone think of a real-life situation in which these angles might arise?)
- The reasonable interpretation of negative angles.
On this last point, I think it's very important to make the analogy with negative numbers explicit. How did mathematicians come to accept the existence of negative numbers? By interpreting positive & negative as different directions on the number line. In exactly the same way, we interpret positive & negative angle measures as rotations in different directions from the positive real axis (MP 2).
With the unit circle firmly established as the key to a new understanding of the trigonometric functions (they aren't ratios anymore), it's time to take the next step: introducing radian measure. I'll begin with the simple question, "We've been talking about degrees quite a bit. Can anyone tell me what a degree is?" My follow-up to this question is "Why are there 360 of them in a full rotation?" (MP 2)
This opening should be enough to launch a very brief overview of the history of degrees, minutes, and seconds. Along the way, I'll emphasize the fact that there isn't anything about the degree that makes it a natural way to measure angles. With this little prelude, I'll introduce the concept of radian measure as the length of the arc on the unit circle subtended by the central angle. The idea is simple: We could just as easily measure how far "around the circle" based on the arc length as on the central angle.
There's a pretty good explanation of this notion here. The main points are:
- Degree measure is used for historical/cultural reasons
- Radians were developed for mathematical reasons
- The radius, diameter, and circumference are the "natural" candidates for a new unit.
- 1 radian is defined as the angle that is subtended by an arc equal to 1 radius
- Example of how to convert from degrees to radians and vice versa
My students should already be very comfortable with other kinds of unit conversions, and this is really no different. The essential fact is that 360 degrees equals 2 pi radians.
Working with Radians
Now that students know what radians are, the only thing left is to gain some familiarity. I'll handout the problem set, Radian Practice, and ask everyone to begin by working individually. As they begin, I'll move around the room checking for understanding and making sure that no one's simply letting their calculator do the conversions for them. If students are giving the radian measure in decimals, I'll ask them to also provide the measure in terms of pi.
Students will continue working on their own until I'm satisfied that everyone knows how to do the conversions in both directions. At that point, I'll let them begin working in small groups if they'd like. These problems are so straightforward, however, that they really should only be using their classmates to check their work.
On problem 3, watch to see who notices that parts b and d are negative. I will not warn the class about these ones; it's more instructive to make the mistake and have to correct it.
Problems 4 - 11 are intended to draw out some of the advantages of radian measure over degree measure. They should see that both arc lengths sector areas are easier to calculate when the angle is given in radians. Problems 10 and 11 lead to the conclusion that when x is close to zero, we can say that sin(x) is approximately equal to x but only when x is measured in radians. This will be vitally important in calculus, but for now it's merely a curiosity. We will discuss all of this tomorrow.
Before class ends, I will review the solutions to problems 1 - 3 of Radian Practice with the class and we will discuss any issues that arise (especially the negative solutions in 3b and 3d). Depending on how far the class has progressed, we may also review the numerical solutions to problems 4, 5, 8, and 9. The rest will be homework and fodder for discussion tomorrow.