##
* *Reflection: High Expectations
The Trigonometric Functions - Section 2: The Sine Function

Isn't it enough just to say that a "periodic function" is one that "repeats itself again and again"? Most normal people can't make heads or tails of a mathematician's definition, so why can't we just stick with something simple?

I like to think of the phrase "a function that repeats itself again and again" as a *naive* definition of a periodic function. It gets at the main idea in terms that are easy to understand, and when it's given along with an example or two, it is an excellent place to start. The problem is that it's too broad. In our inexperience, we have been too quick to generalize.

Students should *begin* with informal definitions like these, but they should gradually refine them to make them more and more rigorous. We need to challenge the naive definition by asking questions like, "What do you mean by repeating? How many times does it have to repeat?" Better yet, we should come up with counter-examples.

Would you call x*sin(x) periodic? What about sin(100/x)? By debating these and other functions, mathematicians gradually refined the definition so that it would include only the functions they intended.

Our students have to experience this process for themselves. While we may not arrive at the "formal" definition that a Ph.D. would use, we should leave the naive & intuitive definitions far behind.

# 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.

## Big Idea: The unit circle allows us to extend the trigonometric functions beyond the confines of a right triangle.

*45 minutes*

#### Advantages of Radian Measure

*10 min*

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!

#### Resources

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#### The Sine Function

*10 min*

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.

#### Resources

*expand content*

#### A Real-World Example

*20 min*

"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.

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#### Wrapping Up

*5 min*

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)!

#### Resources

*expand content*

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: What do Triangles have to do with Circles?
- LESSON 2: Introducing: Radians!
- LESSON 3: The Trigonometric Functions
- LESSON 4: Generalizing the Sine Function
- LESSON 5: Modeling with Periodic Functions
- LESSON 6: Practicing with Sine and Cosine
- LESSON 7: More Modeling with Periodic Functions
- LESSON 8: The Tangent Function