Welcome back, everyone. So at this point, we should be familiar with the graphs for most of the trigonometric functions. We've talked about the sine and cosine graphs, the secant and cosecant. We even looked at the tangent graph in the previous video. Now last but not least, we're going to be taking a look at the cotangent graph.

And this might sound a bit complicated just because learning about another trigonometric graph might feel kind of exhausting at this point, but what we're going to learn is that the cotangent graph is actually closely related to the tangent graph. And if you recognize how these two are related, then sketching this graph is actually pretty straightforward. So without further ado, let's get right into things. Now recall how based on reciprocal identities, the cotangent is just one over the tangent. So we can use this fact to draw the graph for the cotangent.

Now the first thing that I'm going to look for is where we can draw these asymptotes because this will tell us how wide each of the cycles of our graph is. Now if I look at the tangent graph, I can see that we reach 0 at negative pi, 0, and pi. So we reach 0 on the y-axis at all these points. And it is a fundamental math rule that you cannot divide by 0. So any place that we see 0 on the tangent is going to be a value that cannot exist on our cotangent graph, so that's going to be where we draw the asymptotes.

So there's going to be an asymptote right here at negative pi. There's also going to be an asymptote here at 0 on the x-axis, and there's going to be an asymptote over here at positive pi. Now from here, we need to draw the curves. And it turns out that the curves for the cotangent graph are similar to the tangent graph, but they're going to be flipped upside down. So rather than having curves that go from bottom left to top right, we're going to have curves that go from top left to bottom right.

So the graph is going to look something like this. Now I want you to notice all the points on the cotangent graph where we touch the x-axis. This is where our output is going to be equal to 0. And we can actually see that these values make sense when we look at the tangent graph because notice that we touch the x-axis at negative 3 pi over 2. And on our tangent graph, our graph gets very small when we get to negative 3 pi over 2 in the negative direction.

So basically we go to negative infinity. And whenever you divide a number that is approaching infinity, whether it's a negative or positive number, then your entire fraction or output is going to be equal to 0 approximately, or it's going to approach 0. So that is why we have 0 at negative 3 pi over 2, at negative pi over 2, at pi over 2, and at 3 pi over 2. So you can see at all of these points on our tangent graph, we're either getting really big there towards infinity or really small towards negative infinity. Now one of the main similarities between the tangent and cotangent graph is the distance on our x-axis, which we have the curves repeat.

Because notice for our tangent graph that we repeat from negative pi over 2 to pi over 2, which is just a distance of pi. And it is a distance of pi for each of these curves on this graph. So that is one of the nice things about the tangent graph is we repeat every pi units, and it turns out that the cotangent graph is the exact same thing. Because if you look, we can see here our graph goes from negative pi to 0, and that's just the distance of pi. And we also go from 0 to pi, which is also a distance of pi.

So it turns out that this graph repeats every pi amount of units as well. Now because these graphs have the same repeating behavior, it turns out their periods are also going to be the same. So just like the period of the tangent was pi divided by b, the period of the cotangent is also pi divided by b. Now the way these graphs are different from each other is the values of their asymptotes. So for example, if we take a look at the tangent graph, we can see that these asymptotes will always show up at odd multiples of pi over 2.

Notice we have an asymptote at negative 3 pi over 2, negative pi over 2. We have an asymptote at pi over 2, 3 pi over 2. Then if we kept extending this graph, we'd also have asymptotes at 5 pi over 2 and 7 pi over 2. Now notice for our cotangent graph, our asymptotes are at negative pi. They're at 0.

They're at pi. If we kept extending this graph, we would see asymptotes at 2 pi, 3 pi, 4 pi. So it turns out for the cotangent graph, the asymptotes are always going to be at integer multiples of pi. So the location of the asymptotes on the x-axis, as well as the fact that these curves are flipped upside down, are really the two main differences between the cotangent and tangent graphs. Now to make sure we understand this concept well, let's try an example where we have to deal with the cotangent.

Now to solve this problem, what I'm first going to do is figure out where the asymptotes are going to be located on this graph. And I can do this by looking at what our cotangent is. Now recall that the period of the cotangent is just pi divided by b. Now what I can see here is that our b value is the value we have in front of the x, and I can see that b value is equal to pi. So we're going to have pi over pi, and whenever you divide one number by itself, it's just going to be equal to 1, because they're both going to cancel and give you 1.

So that means that our period is equal to 1. So what I can do on our graph is I can draw points every one unit on our x-axis. So I'll go 1, 2, 3 to the right, and they'll go negative 1, negative 2, negative 3 to the left. Now the period of this graph is 1, meaning I can draw these asymptotes for every one unit. So we're going to have an asymptote here at 0, we're going to have it at 1, and then at 2, and then at 3.

And then these asymptotes are also going to repeat behind the graph as well. So I have an asymptote at negative 1, negative 2, negative 3, and this is what the asymptotes are going to look like. Now my last step for solving this problem is going to be to draw the curves. And recall that the curves go from top left to bottom right between these asymptotes. So what I'm going to do is have a curve that looks something like that, and these curves are just going to keep repeating for each period of this graph because that's what the repeating function is going to do.

So this right here is the graph of our function and the solution to this problem. So that is how you can deal with graphs when it comes to the cotangent function. Hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.