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Graphs of the Sine and Cosine Functions - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Sine and cosine are periodic functions, which means that sine and cosine graphs repeat themselves in patterns. You can graph sine and cosine functions by understanding their period and amplitude. Sine and cosine graphs are related to the graph of the tangent function, though the graphs look very different.

I want to talk about graphing the sine and cosine functions. But first, I need to go over a property that the sine and cosine functions have and that these three functions have. So I have the question, what do these functions have in common? I have three, very differently shaped functions, but they all have something in common: They repeat themselves periodically. This triangular shape repeats itself in the graph of y=t(x). This alternating interval pattern repeats itself in y=r(x). And this wave pattern repeats in y=s(x). How would we describe this property mathematically?
Well the property is called periodicity and these functions are called periodic functions. The definition's a little bit tricky, but let's see if we can walk through it and understand what it means. It says, if there is a number P, such that f(x)+P=f(x) for all of x in the domain of f, then f is a periodic function. What does this mean, f(x+P)=f(x)? It means that if I find the right value of P, I can always add that value to the x, to the input and get the same output.
Let's take a look at the functions. Suppose I start with an x value of 2. What could I add to 2 and get exactly the same output that I have here, which is 0? I could add two and that will give me 0. If I add two to 2, I get x equals 0, which has an output of 0. And if I add two to 0, I get 2, and that has an output of 0. If I add two again, I get 4, and that has an output of 0.
So have I found the P value that I need? The answer is no, because that P value won't work for all inputs. Let me give you an example, 3. If I add two to 3, I get 1. And the output of 1 is 1, it's not 1, so I have different outputs. Again, if I start at 1 and I add 2, I get to 1. The output at one is +1, and the output of 1 is 1, different outputs.
So I have to find another P value, one that works for all x's. And it turns out that the value is this difference, 4. I can get from one maximum to another by adding 4: 3+4=1. So four is the number that's going to work. And I would say that t(x)+4=t(x).
Let's take a look at y=r(x). Now here, if I started a nice number like 2 and I add 2, I do get the same output. Here the output is 0, and here the output's 0. And if I add two again, the output is still 0. But now, I'm a little skeptical; I want to try this out for other inputs. So let me try it out for something like 1.5. And for 1.5, the output is 1. If I add two to that I get 0.5, and there the output is 1. And if I add two to that I get 2.5, and there the output is 1.

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