The Trigonometric Functions
Lesson 3 of 8
Objective: SWBAT interpret the trigonometric functions in terms of the unit circle. SWBAT graph a sinusoidal function. SWBAT use a sinusoidal function to make interpretations in a modeling context.
Advantages of Radian Measure
I'll kick off class with the question, "What is a radian?" The class should be able to provide a good definition and explain how to convert from degrees to radians, and I'll do my best to call on some of the quieter students for this.
With this very brief refresher, it's time to discuss the solutions to problems 4 - 11 from Radian Practice. I'll have one student come to the board to demonstrate her solution to problem 5, and then have another demonstrate the solution to problem 7. Clearly, arc lengths are easier to calculate if the angle is given in radians!
Next, we'll turn to the area of a sector. Although it isn't given in general form in the problem set, I'll ask a student to come to the board to explain in general how sector area is calculated when the angle's given in degrees. We'll compare this to the general formula in problem 9 to see that, once again, radians have the advantage! In essence, by incorporating the factor, pi, into the angle measure, we can eliminate it from the other formulas.
Finally, I'll ask students what they noticed about the table they were asked to complete. Since they are not used to thinking in terms of limits, they may not have noticed much. In this case, I'll ask, "What would you expect to see if the radian measure continues to get closer to zero?" The answer is that the ratio should get closer and closer to 1. "Is this true when the angle is measured in degrees?" No. Once again, radians are nicer than degrees! This particular fact won't have much use in Algebra 2, but it is fundamental to calculus and simply interesting in its own right. (By the way, when the angle is measured in degrees, the ratio approaches pi/180. Pretty neat!)
The moral of the story: Radians are preferred by mathematicians!
The Sine Function
I'll begin the next section by asking, "Do any of you recall the definition of function in mathematics?" Once we've recalled the essential characteristics of a function, I'll ask, "Can we say that sine and cosine are functions?" To help students see that the answer is yes, we'll briefly consider the following:
- What quantities are the "inputs" for the sine function? angles or arcs
- What quantities are the "outputs" for the sine function? ratios or lengths in the unit circle
- Is there always just one output for each input? Yes
- What's the domain? All real numbers.
- What's the range? Real numbers from -1 to 1.
Once we've reached this point, I'll take some time to discuss the graphs of the sine and cosine functions. There are many great applets and animations on the web that make the graphs easier to understand (I personally like this one). Using the animation, we'll discuss the key features of the graph and how they relate to the unit circle. Of course, I'll pay special attention to the periodicity of the Sine_Function. I'll need to define periodic function for them, too.
A Real-World Example
"What are some good examples of periodic phenomena in the real world?" (MP 4)
I'll ask my class this question to see what they come up with. I'm expecting things like these:
- Planetary orbits, seasons, tides, phases of the moon
- Sound waves, water waves, light waves, radio waves, and other waves
- A rolling wheel, a mass bouncing on a spring, a swinging pendulum
These are all great examples, but for our first modeling problem we'll begin with something a little different: electricity. Alternating current is an example of a real-life periodic phenomenon. The current (and voltage) fluctuate periodically and may be modeled nicely with a fairly simple sine function.
[N.B. We have not yet discussed amplitude, period, frequency, etc. This problem will serve as a motivation for all of that.]
I'll hand out the AC Generator Problem and ask my students to take a few minutes to read the beginning. Then after an opportunity to ask questions about the physics behind the problem, I'll let them begin working in small groups to complete the graph. As they work, I'll move around the classroom helping individuals and checking the students work so that I can catch & correct their mistakes quickly.
The key is for students to take a systematic approach. Please see this video for details.
We've covered a lot of ground today, and I've had many opportunities to check for understanding along the way. My expectation is that by the end of class, my students will be finished with parts (a) through (d) of the AC Generator Problem. For homework, I will likely assign part (e) only.
Developing the equation for current from the voltage equation is a good exercise, and this minor homework assignment will give students who need it the chance to catch up. Also, in case students make mistakes in part (e), I would hate for them to graph the wrong equation it part (f)!