 In this video I want to talk about the graphs of secant and cosecant. In previous videos we talked extensively about the graphs of sine and cosine as well as tangent and cotangent. It turns out because secant and cosecant are the reciprocals of cosine and sine, our knowledge of graphing sine and cosine will help us graph secant and cosecant. So I would say that graphing secant and cosecant is a lot easier than graphing tangent and cotangent because we are more familiar with graphing sine and cosine. So I'll explain that in just a second. So you see the graph of secant presently on the screen illustrated here in yellow. Now cosecant for the most part its graph looks like these buckets. Every once in a while you have a bucket that goes up, every once in a while you have a bucket that goes down. The proper word we'd say is that this portion of the graph is concave upward and this portion of the graph is concave downward. Cosine is a two pi periodic function and secant, which is of course the reciprocal of cosine, it'll inherit that same period. So the period for secant is going to always be two pi as well. And so that will be important for the standard graph. It'll also be important as we repeat the graph. Now you'll notice that when you look at a bucket this bucket goes from negative pi to negative pi halves to pi halves right there. And then the next bucket goes from pi halves to three pi halves. So it's tempting to be like well wouldn't the period be pi because the bucket repeats itself every pi. That's not true. You have an up bucket and a down bucket, right? And so you don't get back an up bucket until you've done a two pi interval. So two pi is in fact the correct period for this function. Another thing to point out is like tangent and cotangent, the graph of secant is, it has vertical asymptotes. That is, there are places in the domain of secant that you're undefined. And the location of the vertical asymptotes of secant are identical to the location of the vertical asymptotes for tangent. And that's because tangent with sine over cosine, secant is one over cosine. Both of these functions have cosine as their denominator. That's actually why their BFFs right here best spends forever because they're the same common denominator. One consequence of that is that secant and tangent have the exact same vertical asymptotes. So secant will be undefined whenever cosine is equal to zero, which happens at pi halves, three pi halves, five pi halves, negative pi halves, negative three pi halves. So any odd multiple of pi halves, that will be the vertical asymptotes of secant because that's when cosine is equal to zero. All right? When it comes to graphing trigonometric functions, I've always mentioned there's like five special points you want to use. You can always use more than that. The more refined you make your graph, the better. But when it comes to cosine, the standard point you would use is you'd use x equals zero, x equals pi halves, x equals pi, x equals three pi halves, and x equals two pi, which would be over here. I've moved things a little bit. So instead of using the last point of pi, I'm actually going to use negative pi halves. And that way it gives us two complete buckets of the graph. But you could also go from zero to two pi like we did before. That's fine. Now you'll notice on the screen, this green curve is actually cosine right here. And I wanted to include this to show you the relationship between secant and cosine given that the reciprocals of each other. So we're pretty good hopefully by now of graphing cosine and sine waves. You can see the standard cosine wave right here. Because secant is the reciprocal of cosine, their graphs are in some essence opposites of each other. When cosine is small, secant will be big. And if cosine ever got big, secant would then get small. Now, cosine's bounded between one and negative one, which actually means secant avoids one and negative one. It turns out that the absolute value of secant is always greater than or equal to one. An up bucket means you're going to be greater than or equal to one. A down bucket means you're less than or equal to negative one. But the absolute value is always greater than or equal to one because you're avoiding the range of cosine right there. Okay. And so if you draw your standard cosine curve, then it turns out that the secant function will actually wrap away from where you see cosine. So in particular, cosine has x intercepts at negative pi halves, at pi halves, and at three pi halves amongst other points, of course. These are locations of vertical asymptotes because cosine is zero at these points. Secant should be one over zero, which essentially is an explosion. It goes off towards infinity and negative infinity. That's the idea of a vertical asymptote. You're trying to divide by zero and it causes this rip in the space-time continuum or just the function, I guess. Maybe it's not as traumatic there. But then also when you look at the graph here, when you see that this portion of the cosine function where it's concave downward, right, it has this maximum value, this is the point where secant and cosine are going to kiss each other, where you're going to have this point of intersection where you have this point zero common one is common to both secant and cosine. You'll have another kissing point right here at this point right here where you're going to have pi common negative one. So where cosine is negative one, secant will also be negative one. They're going to touch each other, just going to kiss each other. But other than these kissing points, the functions wrap away from each other. So while cosine over here between negative pi halves and pi halves is concave down and curves downward, secant is going to curve upward, forming this up bucket that we talked about before. And likewise, as when you're on the down bucket here, when cosine is concave upward, it curves upward, secant will have this down bucket going on like this. And so using the graph of cosine, you can actually graph secant as well. You're just going to graph it so it wraps away from the cosine function. We can see this if we move over to Desmos, of course. You now currently see on the screen the graph of secant like we saw before. Let me add the asymptotes. These are going to be the places where secant is undefined. You see this up bucket shape that we saw earlier and this down bucket shape as well. So it repeats itself up, down, up, down in this alternating fashion. If I put the graph of cosine on the screen, you see the behavior we talked about before, they kiss whenever cosine has a maximum or a minimum. So it kisses at 0, 1, at pi negative 1, at 2 pi 1, and negative pi negative 1, at negative 2 pi comma 1, you get it. Whenever there is an x intercept of cosine, that's where you're going to have a vertical asymptote for secant. And so the behavior is exactly reciprocal of each other. Like so, whenever cosine is bending downward, secant will bend upward. Whenever cosine is bending upward, secant will bend downward. And so with that knowledge, we can very easily graph a secant function by graphing the cosine function that corresponds to it, and then we just reciprocate. We can have a similar discussion about cosecant because the thing about cosecant is that cosecant of x is the reciprocal of sine. And so because of this, there's this reciprocity relationship between cosecant and sine. If we could graph sine, then cosecant will look the same as secant in so much that we're going to wrap away from sine. So some things to note, sine is undefined at 0, at pi, at 2 pi, at negative pi, at negative 2 pi, and at any multiple of pi, sine is 0. This is going to exactly be the vertical asymptotes of cosecant, which are also the vertical asymptotes of cotangent. Cotangent and cosecant are BFFs because they have the same denominator of sine. So we get vertical asymptotes whenever sine goes to 0. Whenever sine has a maximum, cosecant is going to kiss it at that location. So we have this point right here where the sine's at its maximum. This is going to be pi halves comma 1. You're also going to have kissing occurring whenever sine has a minimum, like for example, at 3 pi halves comma negative 1. So whenever sine has its max or min, you're going to have a kissing relationship between cosecant and sine. Also, when you see curvature with sine, cosecant will have the opposite curvature. So for sine here, we see that it's concave down, it's bending down, the bowl goes down. Therefore, cosecant will have an upward bucket when sine is wrapping down. On the other hand, when sine is curving upward, you have this concave upward shape that you'll get a down bucket for cosecant right here. Some other things to note about cosecant, it's its period is going to equal to 2 pi just like sine. It doesn't have any x-intercepts. In fact, if you look at the range of cosecant, the cosecant gets all real numbers, other than the interval from negative 1 to 1. So in particular, the absolute value of cosecant is greater than equal to 1. Up buckets are going to be greater than equal to 1, and the down bucket is going to be less than or equal to 1. So with these graphs down in mind, we could start graphing modified secants and cosecants. Let's look at a secant graph right here. You have y equals 1 third secant of 2x. We can start graphing this by transformations. There's a vertical compression of 3 and a horizontal compression of 2 when we graph that. But instead, when it comes to graphing secant and cosecant, my suggestion is actually we're going to graph first the function y equals 1 third cosine of 2x. We're more comfortable graphing sine and cosine, so let's do that. If we graph cosine, we're going to have an amplitude of 1 third, and we're going to have a period of 2 pi over 2. That is a period of pi. And so using that, we can then graph our five points that we would normally want to do. So we can go from 0 to pi if we wanted to. I'm going to take one step to the left and actually start at negative pi force. So negative pi fourth is right here. Zero, we're going to take pi fourth, pi halves, three pi halves, excuse me, three pi fourths like so. And so when it comes to graphing a trigonometric function, apparently every picture looks the same. It just comes up to labeling. If you relabel the graph, or I should say if you leave the grid lines blank, all the pictures look the same. So really the hardest part about graphing a trigonometric function is making sure you have the right intervals, all right? Since the amplitude is going to be one third, I'm actually going to put this right here as one third. So this is two thirds, and then this is going to be one. And then it's going to be negative one third, negative two thirds, and one and below as well. So we're going to put the start and ending of a single period here. So we're going to go from negative pi fourths to three pi fourths. That is a length of pi. Like I said before, we could have gone from zero to two pi. That would be acceptable. But because of the bucket shape of secant, it's going to be a little bit better if we start at negative pi halves, negative pi fourths, excuse me, because if we were to draw the cosine function, we're going to get something like the following, right? So that's one complete cycle of cosine. It kind of looks like a sign because we started to the left, and that's okay. So the thing to note now is that when it comes to our secant function, we have vertical asymptotes occurring at the x intercepts of cosine. So we're going to have a vertical asymptote at negative pi fourths. So this is why we graph cosine first, because from the cosine function, we can very easily construct our secant function. So we have a vertical asymptote at negative pi fourths at pi fourths and at three pi fourths, like so. And so now we're going to draw our function so it wraps away from the cosine. So they will touch at their, they're going to kiss at the maximum. So this should be maximum at zero and a minimum at pi halves. Otherwise, we're going to wrap away going towards the vertical asymptotes, like so. There's our up bucket. And then we got our down bucket as well, right here. And so then we can graph our function y equals one third secant of 2x. And so that's going to be my strategy for you. If you have to graph secant or cosecant, first graph sine or cosine as a helper function and then reciprocate in order to get the graph of secant or cosecant. And so that's going to end lecture 12 here about the graphs of tangent, cotangent, and cosecant, and cosecant. It also really ends our discussion of graphing trigonometric functions. We do have one more unit in chapter four about graphing trigonometric functions. That's going to be applications in particular. We're going to talk about simple harmonic motion. Check the link you can see on the screen right now to watch that video. It's going to be a good one. Thanks for watching, everyone. If you want to see more videos like this, of course, subscribe. And always, if you have questions, post them in the comments below, and I'll be happy to answer them. Bye, everyone.