- Cosecant of x-- we'll just start with the basic one. We know that cosecant is the reciprocal of the sine graph. So what we're going to do is we're going to graph the sine graph to use it as a guide. So if we graph the sine graph, we know it looks like this. And it keeps going and so on, like that. 

But if we look at the sine graph here, say where sine is equal to 0. What's 1 divided by 0? It's undefined, right? So what happens is you get a vertical asymptote. OK. Same thing here at 0-- what's 1 divided by 0? It's undefined. You get a vertical asymptote. So wherever the graph, the sine graph is crossing the x-axis, that's where your vertical asymptotes are going to go. 

Now, over here, with sine, the parent function for sine goes between negative 1 and 1. Over here, see, sine is 1. What's 1 divided by 1? It's 1, right? Say over here it's 1/2. What's 1 divided by 1/2? That's 2. OK. So you can see 1/2, 2. So what happens is we get a graph that looks like this and like this. 

So we just graphed the sine graph to help us OK. We use it as a guide to get our graph for cosecant. OK. So over here, this is-- let's see. This is pi. This is 2 pi, 3 pi, like that. So that's the basic graph for cosecant. 

Now, for cosine, it's the exact same idea. So say, for example, we want to graph secant. Secant is the reciprocal of cosine. So what we're going to do is we're going to graph the cosine graph, which starts at the maximum, goes to 0, down to the minimum, back to 0, back to the max, like that. And, of course, repeats, so we can keep going, like this. Just draw a little bit more than one cycle. 

And then, again, wherever it crosses the x-axis, because this is 0-- we can't divide by 0-- we're going to get vertical asymptotes. So there's your vertical asymptotes. And again, we can label this. This is a pi over 2. This is pi, 3 pi over 2, 2 pi, and so on. And then our graph is going to coincide at this maximum point and at this minimum point. They coincide. They just cross right at that one point. 

So a quick way to get your basic cosecant and secant graph by graphing the cosine or sine and then taking the reciprocal, like so. OK. Let me show you a little bit more complicated example. S You can get a sense of how to do some of the more difficult problems. Let me just erase this real quick. 

So say we want to graph y equals 2 cosecant 1/2 x. It's a pretty good problem, right? What we're going to do is we're going to graph y equals 2 sine 1/2 x. So what does that mean? It means the amplitude is going to be 2. We're going to graph our sine graph, like that. I'm going to graph a little bit more, just to give us a little bit better diagram here, a better picture. 

And remember, the period is 2 pi-- is the normal period-- divided by b. This is your b value here, 1/2. So that comes out to 4 pi. So what that means is it's completing one cycle, one period, in 4 pi OK. So that means this would be 2 pi, 1 pi, 3 pi. So I'm just dividing this into four pieces. 

OK. Now we can go ahead and graph our cosecant graph. Just wherever it crosses the x-axis, we're drawing our asymptotes, like so, wherever the sine graph is crossing the x-axis. And then we can just draw our cosecant graph. And that's it. You got it. 

So I'll show you one more example with secant. We're going to do a little bit more challenging one. Always have to step it up a notch, right? Start easy and then build on that. Let's do one where it's been translated. So let's look at this one. Let's say we want to graph y equals secant 2x minus pi over 2 plus 1. 

So what does the pi/2 do? What does the 1 do? This one, remember, has the opposite effect. It's shifting right pi/2. This one has the same effect on the graph. It's shifting up 1. So what I'm going to do is I'm going to start by graphing the cosine graph. 

Now follow me here. I'm going to graph the cosine graph just using this graph here. So cosine looks like this. If you need to review back, go back to the cosine and sine videos if you need some help with this. But it starts at the max, 0 min, 0 max. That's one cycle, 2 pi. This is pi/2, pi, 3 pi over 2, like that. 

And the minus pi over 2, this is shifting it actually positive pi over 2, right pi over 2. So each of these points is going to go to the right 1 step, like that, like that, like that, like that. And this point would be right here. So now your graph looks like this. 

All right. So, so far, with me? OK. But now, remember, we said that wherever it crosses the x-axis, that's where the vertical asymptotes are going to be. So right here, that's where our vertical asymptotes are. And now what I'm going to do is, I'm just going to lightly draw in the secant graph like this. 

And this keeps, going by the way. So I can draw arrows on here. So it keeps going. It approaches the asymptote but doesn't touch, doesn't cross. But wait a second. You say, what about the 1? What do we do with the 1? Well, the 1 tells us to shift the graph up 1. So all these points are going to go up 1. 

So now the graph is going to look like this. And then same thing over here-- this is going to go up 1. So it's actually touching right there, up 1, like that. And the asymptotes are in the same place. So that's how I would do it. I would start graphing this cosine or sine graph, depending if you're using secant or cosecant. And then I would do the shift. This is the phase shift left and right. Wherever it crosses the x-axis, draw your vertical asymptotes. 

Go ahead and draw on your secant graph. You can draw it dotted or in a different color or whatever you prefer. And then you can shift it up or down. So that's what I would recommend. That's the secant graph there. If you have other questions, go back and review the sine and cosine graphs if you have questions on how to get the graphs and the period. The period for secant and cosecant is the same as for sine and cosine. It's 2 pi. So whatever this number is right here in front of the x variable, you can use that formula, period equals 2 pi divided by b. This is the b. In this case, it's 1 to find out how long it takes for it to complete one cycle. 

OK. I'll see you in the next video. If you have comments, or questions, or things that you'd like me to discuss more, or helpful hints that I can use to incorporate into these videos, feel free to leave me a comment or send me an email or let me know. And I'll see you in the next video.