## 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

# The Trigonometric Functions

Unit 9: Trigonometric Functions
Lesson 3 of 8

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

Print Lesson
6 teachers like this lesson
Standards:
Subject(s):
Math, Trigonometric functions, Trigonometry, periodic functions, Algebra, master teacher project, sinusoid, radian measure
45 minutes

### Jacob Nazeck

##### Similar Lessons

###### Riding a Ferris Wheel - Day 2 of 2
12th Grade Math » Trigonometric Functions
Big Idea: Make the transition from the Ferris wheel problem to the unit circle.
Favorites(3)
Resources(10)
Troy, MI
Environment: Suburban

12th Grade Math » Rotations and Cyclical Functions
Big Idea: Students use cylinders and string to investigate radian angle measurements and then use their findings to develop a method to convert from radian to degrees.
Favorites(7)
Resources(19)
Phoenix, AZ
Environment: Urban

###### Arc Length and Sector Area
12th Grade Math » Trigonometry: Circles
Big Idea: Students find their friend similarity lurking in a table of repeated calculations. A firm concept of radian measure follows close behind.
Favorites(11)
Resources(14)
Worcester, MA
Environment: Urban