Sine and Cosine Graphs

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.