Trigonometry: Fancy Graphing: TAN, SEC, CSC, COT

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Fancy Graphing: Tangent, Secant, Cosecant and Cotangent
Let's take a look at some sort of real fancy graphing where, in fact, we have coefficients all over the place, inside the x's and then outside, and so on. So how about, let's graph the following: y = tangent of x.
Okay, how would I proceed? Well, what I would do first, what I always do, is just start off by graphing tangent. And then compare that graph to the thing I want. So, first I'm going to do something that was not asked of us. Namely, I'm just going to graph the tangent function first. So how does the tangent function go? Well, the tangent function has some asymptotes at , 3, -, and so on. So the tangent functions have the basic shape that looks like this. It has period just and so on. So there's the tangent. That's not the answer. That's just tangent.
Now, how do we get one-half of tangent? What we do is we take every single value that we have here and now we can take half of it. So, what does that do? It sort of makes things a little bit less loose and more sort of sharp, so the actual graph would always be sort of below this here, because every point here I'd actually look at half of it, so it would look more like this. It would sort of go between the asymptote -- this would still be an asymptote, the blue is still an asymptote, but now it's going to go between the asymptote and this orange curve. You can see it's a little sharper, because every point is replaced by half of it. So every point gets sort of cut in half. So when you put in a coefficient in front here, that's less than one, it sort of makes things sharp. If I put in, for example, like , it would be really sharp, really sharp. If I put in something like 5, it becomes even more smoother. So that's that graph.
Let's compare that to the graph of y = 4 tangent of x. I'll use a completely different color there. Let me use brown. This purple, by the way, is y = tan x. That was what we first graphed. Now let's suppose we're asked to graph y = 4 tan x. Well, now, in fact, I'm going to take every value and, instead of going up to the orange, I'm going to multiply by 4. So what happens is, it gets even smoother looking. Now it's still asymptotic. I mean, I don't want you to think -- this is actually going to get really, really close this asymptote, but it's going to get there much slower. Because every point has been multiplied by 4. So, in fact, this brown is the graph of y = 4 tangent x. And you can see that as I multiply by bigger and bigger things, what happens is this starts to sort of really sort of become very, very gentle smooth. If I put in like, you know, , it becomes very sharp, because I'm dropping the values.
Okay, how about this one? Let's think about cosecant for awhile. What if I want to graph y = cosecant of 2x. Well, the first thing I do is say, okay, I don't know anything about cosecant. I know nothing about cosecant and that is the truth. Except, that it equals . So my first step will be to graph sine of 2x and then take a reciprocal somehow. Now, sine of 2x,that actually is going to have a different period. Let's figure out the period of that. Let's sort of do that. Let's come over here and let's ask ourselves, "What's the period?" The period would be 2 divided by that number, which is 2. So I see . So, the period is . So, actually I'm going to have a whole period in just 1 and then 2, I'll have two periods. Okay, let's put some axes here. And the first thing on the graph before I do anything else, is just the sine of 2x function. So that requires me to do a whole period right in here, so the amplitude is still 1, but I have to cut these things in half a little bit now. Let's just put that in real life. So, it's a regular sine function, but with period . So, it's a little more wavy than usual because in 2, I see two periods. Now, to compute the cosecant, which is what I'm after here, it's no problem. I know there are asymptotes wherever this line crosses the x axis, because that's in that case. So I have it here. Now it's going to be a , at now, 3, and a 2. And also a zero. So we have a lot more asymptotes. At the max points at 1, when I take a reciprocal, I'm still at 1 and that sort of helps me put in this curve. So I'm asymptotic here at the -1. Here I'm at 1. So you can see what this looks like. I just flipped it. So, in fact, the purple is the answer. The purple is the graph of y = cosecant of 2x.
What if you wanted to graph y = 2 cosecant of x? What would that look like? Well, let's compare that to this graph. So let's now graph y = 2 cosecant of x. Well, again, what I do is, I first look at the sine. That's just a good old-fashioned sine function. No change in period or amplitude, just have fun with it. Here's 1, here's -1, here's 2, boom, boom, boom, and I first graph the sine. Now, it's not required of us, but I find it really helps me to place the cosecant. Because I know, for example, wherever this is zero, we're going to have asymptotes. So we're going to have an asymptote here, we're going to have an asymptote here, we're going to have an asymptote here, and so forth, it keeps going. Okay, but now, what do I want to do? Now what I'm going to do is I'm going to flip, but I also have to multiply by 2. So what effect does that have? Well, instead of flipping and getting a 1, I now have to multiply that 1 by a 2. So, in fact, now I'm going to jump up to 2. So, in fact, now this actually jumps up to 2. You see that? Because I'm going to take that point. That's 1. Now I take its reciprocal, that's still 1. But I multiply by 2, I jump to 2. And similarly, here, all these points are going to be the exact same process. I'm going to just take the reciprocal, which makes them big, and then I multiply by 2. So, in fact, what I basically see is this kind of graph where there's a little gap now because here at -1, I multiplied by 2 and I get -2. So, in fact, there's a gap between these things because of that shift to 2, they no longer touch at 1 and -1 for the y values, because I multiplied by 2 and I get 2, multiply this by 2 and I get -2. Same picture otherwise. So, in fact, there's a big difference between look at 2 cosecant of x versus cosecant of 2x. Cosecant of 2x puts in a lot more terms because the period changes, but they still touch, whereas 2 cosecant x, we just have one period and 2, but now these things go way out. So it changes things dramatically.
All right, let's try one last one. We sort of put a whole bunch of stuff together. Let's see if we can sketch the following: y = 3 secant of x. So, again, my first thought is to convert this to 3 times 1 over cosine x. There's the period issue here with the period. The period is going to be 2 divided by this coefficient, which is , which equals 4. So I've got to wait now 4 until I get one complete period. This would really be dragged out. I take that, take a reciprocal and multiply that answer by 3. So what does that look like? I have to go way out, to 4. So, here's 2, as usual, but now I have to go to 4. This is 3, this is . So I have to do a whole period right here. I have to really stretch out that period. So let's do cosine. Let me put in 1 here, by the way. Let's suppose here is 1. So, here's cosine. Start up at 1 and I come down, very gradual. Just go to 1. This is supposed to be -1 here. Perfect. All right, now what I want to do is I want to look at the reciprocal. So, in fact, I'm going to have asymptotes wherever this is zero. So I have an asymptote here at , an asymptote here at 3, and then what happens? I take this value, I'm going to flip it, but then multiply it by 3. So what is that going to do? Well, in fact that's going to sort of elongate things dramatically. Because instead of being 1, I'm going to be a 3. And sort of here is -2, -3. So, what I see is, the function sort of starts up here and goes. The function starts way over here and goes down. The function starts way up here at 3. You can't even see it. In fact, wait a minute, I better just remove myself from the picture here. Watch this. I'm going to remove myself. I always love it when I can do this. I'm just going to take my hand and just remove myself. Watch this. You're going to love this. There you go. Now, what I can do is show you exactly how this goes. This 1, when we take the reciprocal, it's 1, but then I multiple by 3, it goes way up here and so I get this kind of picture. And, of course, it keeps going. So, you can see that 3 times the secant of x, I've got to wait all the way to 4 for a complete period and then what does it look like? These things are way high up and their low points are at 3 here, and their high points here are happening at -3 and you can see how these things go. The asymptotes are plain. It's wherever the cosine of x = 0.
Okay, so you can start graphing even really, really exotic things such as these functions, by just graphing sort of the basic core function and, if you're taking an inverse, you know reciprocal, take a reciprocal, be careful of multiplicative things and seeing what happens if you sort of take a 1 and take a reciprocal and multiply it by 3, it jumps way up to here. And then, you're home free.
All right, try these and see how you make out.