Unit
Trigonometric Functions
Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
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Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
I want to review what we’ve talked about, about the graphs of sine and cosine. So let’s take a look at a demonstration in Geomer’s sketch pad. We will look at the two graphs together and how they come from the unit circle.
Here we are on Geomer’s sketch pad, and in this construction I’ve got the unit circle drawn down here. And here’s my angle. This is the initial side this is the terminal side. And here is point p whose coordinates define the cosine and the sine of angle theta, right now theta is 1. So I want to show you that I can rotate the terminal side, so you can see that I’ve also plotted the angle theta on this x axis this horizontal axis. Because what I want to do is want to show the graph of sine and cosine.
So let me start by showing cosine. Remember the, cosine is the first coordinate of this point, which is also this length, this horizontal length in my construction. And I have that same length plotted up here on my graph. Now let me rotate the terminal side and what you are seeing here is the graph of cosine.
As the angle rotates around the circle you get the graph of cosine drawn out for you. Let me stop that and let me show you sine. Sine comes from the second coordinate of this point that’s the same as this vertical length that I’ve drawn in red. And I have the same length up here on the graph. I let the angle rotate, now you see a graph of the sine function.
What I know to show you here is that if I graph both of them at the same time, you can see that the two graphs actually have the same shape. One is just a horizontal shift of the other. The orange graph is the cosine graph, the red graph is the sine graph. Both graphs shave a period of 2 pi.
I’m only showing you one period here and of course they go on forever on both directions, these two graphs. But I just wanted to give you the idea, the shapes are really exactly the same. Now in a moment I’m going to draw all these graphs on the same board, and review what we’ve discussed here.
So here are the graphs of sine and cosine. Now on the demonstration we just looked at. I showed you this much of the graph, but sine and cosine are defined for all real numbers. So the graphs go on forever. I’m showing you two full periods of the graphs of and cosine.
Now I want to talk about what characteristics these two functions have in common. First of all, as I just said, they are both defined for all real numbers. And that means their domain is all real numbers. What about the range? They both have exactly the same range because they both have a maximum value of 1, a minimum value of negative 1, and they had everything in between so the range for both functions, is the interval of 1 to 1.
Amplitude, remember this is a term that we use when we are talking about periodic functions. Amplitude has to do with the maximum and minimum values also. And because the maximum and minimum value are 1 and negative 1, the amplitude is 1 minus negative 1 over 2.
The maximum value minus the minimum value, over 2 that’s the amplitude. And that’s 2 over 2, which is 1 both have an amplitude of 1. And the period is the length of time it takes to go through one complete cycle. And after that they keep repeating that cycle.
The sine graph in red here for example, here is one complete cycle of the sine graph, and then it repeats itself exactly. Here’s one complete cycle of cosine and then it repeats itself exactly. They both have exactly the same period, 2pi. So I’m going to put that up.
Now there have some things different about them their x intercepts are different. The sine function intercepts the x axis at 0 pi, 2pi, 3pi and 4pi. Now those are the integer multiplies of pi. So I would say 0 pi, 2pi and of course this goes in both directions. So and then the x intercepts of cosine are pi over 2, 3pi over 2, 5pi over 2, 7pi over 2. These are the add multiples of pi over 2, but just to get an idea of the pattern, pi over 2, 3pi over 2, 5pi over 2, is all I'm going to write.
Now what’s interesting is, the turning points are kind of related to the intercepts. The intercepts of one graph are the running points of the other. For example the intercepts of sine pi 0, 2pi are the running points of cosine and vice versa. So we just switch turning points for a sine pi over 2, 3pi over 2 , 5pi over 2 and the turning points for cosine are the integer multiples of pi.
0 pi, 2pi 7. And of course one of the beautiful things about the sine cosine functions, is that they have exactly the same shape each is a translation of the other. If you shifted the cosine graph pi over 2 units to the right, you get the sine graph.
So they have the same shape, they have the same domain, same range, same amplitude, and period but there x intercepts and turning points are the reverse in each other. That’s the sine and cosine graph.