INSTRUCTOR: Hello, and welcome to My Secret Math Tutor. For this video, we're going to work on graphing tangent and cotangent. So these two are very interesting trigonometric functions. And when you look at them, they don't quite look like your normal sine and cosine functions because, of course, they're not. They are tangent and cotangent. 

The key for really graphing these is recognizing the sine and cosine that work in the background. And you'll see in the process that it's usually a good idea to start with the asymptotes and then find some key values around those. So let's go ahead and get this process started and see how we could build the graph of tangent and some key things we want to put in there. 

All right, so this first one we want to graph is tangent. And like I mentioned before, the key is really recognizing that tangent is sine over cosine. Now, that's important to recognize because it gives you a clue on where to put the asymptotes. 

The asymptotes will happen wherever the bottom is equal to 0. So you want to think back to what the graph of cosine looks like and essentially put an asymptote everywhere cosine would equal 0. So let's go ahead and grab our rulers and start with that. 

So I'm going to put a few tick marks here. Let's do every 2 inches. And then let's mark out what these little tick marks represent. So cosine is equal to 0 at pi over 2. Then it's equal to 0 again at 3 pi over 2. 

And if you want to go in the negative direction, heading off that way, then you would have negative pi over 2 and negative 3 pi over 2. So, again, I'm getting these guys by figuring out where cosine is equal to 0. 

Now that we have those, let's go ahead and mark out the asymptotes by really just making a dotted line, a nice vertical dotted line. And we'll do this for each of our key points we built here, so nice dotted line here. One here. And one more, negative 3 pi over 2. 

So we have our asymptotes. Now what do we do? Well, the graph of tangent essentially wants to hug really close to these asymptotes. And then it actually equals 0 at the halfway point between them. So that means when I'm halfway between these two asymptotes, it'll equal 0. When I'm halfway between these two, it'll equal 0 again. And when I'm halfway between these two, 0. 

And this has the shape of a cubic function. So imagine hugging really close to this asymptote and then eventually coming in, going through that point, and then hugging close to the other asymptote. And it'll just keep getting closer to those asymptotes. So this would be one period of tangent, and we'd simply repeat this for the rest of the graph. 

So I'm going to go through that little 0 right there and then continue getting close to the asymptotes. So there's another period. And let's go ahead and draw one more. Of course, it'd continue like this infinitely in both directions, always getting closer to these asymptotes when it reaches one but never quite touching it. Cool. 

All right, so since we see it has these zer0 points, it's probably a good idea to mark out where these are as well. So the first zero point happens at 0. Then you want to think of the halfway point between these two values, so pi over 2, 2 pi over 2, 3 pi over 2, so this is just pi. 

And, of course, going in the other direction, we'd have negative pi. So all good places where this thing would equal 0. 

All right, moving on. Sometimes you may wonder how far up and down this should be stretched. And it's really tough to tell because usually with sine or cosine, we can have the amplitude marked out by its highest and lowest peak, whereas this one it keeps going off to infinity, keeps going down to negative infinity, but there are some key points that you can mark out that will help you figure out the stretching factor when you get to transformations. 

And that's these two key points right here at 1 and negative 1. Essentially, that's the halfway point between the 0 and the asymptote. And it's another key point where it goes through the graph. So we'll mark out these as well. Let's see. 

So halfway between these two, I'd be down at negative 1. Halfway between these, I'm up here at 1. Halfway at 1 and, let's see, halfway at negative 1. So let's go ahead and mark these out so we can figure out where they're at. 

So halfway between these two, we are at pi over 4. Halfway between pi over 2 and just pi, let's see, we got to count in fourths. So we have 1/4, 2/4 3/4 pi. 3/4, 4/4, 5/4. And the same values would be in the other direction, so negative pi over 4, negative 3 pi over 4, and negative 5 pi over 4. Nice. 

So that's our basic graph of tangent. And if you had to build a basic one, you want to really follow it in the same way. So start off with your asymptotes by figuring out where the bottom is equal to 0. That's cosine. And then go ahead and mark out your zeros. 

Those are the halfway points between your asymptotes. That way, you can actually draw in your graphs. And if you want some other additional points in here, then go ahead and chop it in half one more time and you'll figure out where the graph goes through 1 and negative 1. And you'll have a really good picture of tangent. 

All right, let's go ahead and move on and do the same building process for cotangent. So cotangent you can build in a very similar way. You would first start off and recognize that cotangent is really cosine over sine. So we're going to put our asymptotes everywhere that the bottom is equal to 0, so everywhere where sine is equal to 0. 

All right, let's go ahead and grab our ruler and start making out some little tick marks here. One of the first places that sine is equal to 0 is actually right on the y-axis. So that'd be one of our asymptotes right there. We'll go 2 inches out. We won't go all the way out. The 2 inches would be right about there. 2 inches this way, another 2 inches. OK, good. 

So these guys, if we had to represent them, are at 0. And the first place where sine is equal to 0 is right at pi. And the next time way out here would be at 2 pi. And the same for the negative direction, so negative pi, negative 2 pi. Looks good. All right, so let's draw in those asymptotes. 

I'm just going to draw a portion of one over here. I don't want to get in the way of what I've already wrote. Let's see. Let's get another good one right there. We'll still be able to see two periods of this since our y-axis happens to be one of our asymptotes. In fact, let's go ahead and mark that out. Even though there is a line already there, we do want to remember that this is an asymptote as well. Let's go ahead. 

All right, so we got some good asymptotes on here. I know that my graph will follow them. Now let's start chopping each of these in half to see where it goes through at 0. So halfway between 0 and pi is at pi over 2. 

Halfway between pi and 2 pi, this would be 3 pi over 2. So 1/2, 2/2, 3/2, 4/2, looks good. Going into the negative direction, negative pi over 2. Halfway, negative 3 pi over 2. Nice. 

Now the graph of cotangent looks an awful lot like the graph of tangent. But you want to draw your curves going the other way. So tangent would start off at this right asymptote and then start hugging towards this way a lot like x cubed. 

Cotangent works the other way. So we're going to start hugging really close to this asymptote. Then we're going to come down and go right through our 0 and then hug close to our other asymptote. And there we would have one period of cotangent. 

Then we just want to repeat this process. So here would be another one. Go through our 0 point, hug close to the asymptote, and let's draw one more. 

So let's see. So this is coming down, going through our 0 and then hugging close to the asymptote. And, of course, this would just repeat. We'd have just one little bit more over here. And that would be our graph of cotangent. 

Now like tangent, we can also mark out some other key values on this, places where it would go through 1 and negative 1. And these places would happen by chopping the intervals in half one more time. So you're exactly halfway between 0 and pi over 2. 

So let's go ahead and mark out the first one. This would be at pi over 4. And it reaches negative 1 at 1/4, 2/4, 3/4 pi, negative 1. Come out a little bit further, and let's go ahead and chop this in half. So we're at 1-- oh, it looks like I didn't draw my graph entirely accurate-- and negative 1. 

So let's see, 4/4, 5/4 pi. 6/4, 7/4 pi. And these would be the same values going in the negative direction. So halfway at negative pi over 4, 1/4, 2/4. Halfway at 3 pi over 4 negative. So we're up here at 1, down there at negative 1. 3/4, 4/4, 5/4. And let's make that our last one. Neat. 

So very similar. But think of the same process when you're building this one. First, find your asymptotes wherever sine is equal to 0. Then go ahead and figure out where the function is going to be 0. That will actually happen wherever the top is equal to 0. Or you can end up just chopping these in half. That's what I like to do. 

Then you can go ahead and draw out your graph, making sure that for cotangent it's actually flipped the other way. And then lastly, go ahead and add in a few more key points. These ones at 1 and negative 1 will give you just a little bit more accurate picture. 

Perfect. All right, if you want to make this graphing process even better, it's a really good idea to know the graph of cosine and sine like the back of your hand. So if you haven't already checked out my other video on sine and cosine, go check that other one out. That way, you can learn how to build that one quickly and easily. If you'd like to see some more videos, please visit mysecretmathtutor.com.