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.

  High Expectations: Degrees of Rigor
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The Trigonometric Functions

Unit 9: 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.

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Math, Trigonometric functions, Trigonometry, periodic functions, Algebra, master teacher project, sinusoid, radian measure
  45 minutes
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