PC 10.3 Reciprocal Trig Graphs

Hello and welcome back to another lesson in pre calculus. This is Mister Bean and today we're going to learn about reciprocal triggers. This lesson will be pretty quick, so be ready to go. We're going to first recall the graphs of cosine sine and tangent. Just remember these, the cosine starts off with a maximum, sine is going through the origin here. And then tangent is this weird one where we have all of these vertical asymptotes going up and down. Now, to do the reciprocal graphs, that is when we have, we call them reciprocal reciprocal graphs because it is the trig functions where we take secant X is the same thing as one over cosine. So if you know that it's one over cosine, then you have to think about what would happen if the denominator were to equal zero. If you think back to our rational functions, what happens when we get this denominator equaling zero. There is a discontinuity, so it is either that we have a hole in the graph, something like this, and then the graph continues on, or we would have a vertical asymptote straight up and down. In this case, it would be a vertical asymptote. So let me show you here. You're going to look at all of these places where cosine is zero. That's why I have the graph here. So cosine is zero. Right there. There, there. And there. There's the places where the cosine graph is a zero. And that means that we would have vertical asymptotes on the graph of secant. So I'm going to put some vertical asymptotes, and you'll do this on yours as well. Vertical asymptotes right here at those zero marks. And mister Brussels walk in the door right now, saying hello. He's saying hi in this video. You want to say something about triggers while I'm drawing this? I love trick. Yeah, he loves it. Let's see. So we have our vertical asymptotes. And then from there, what does this graph going to be doing? So here's what we could do. We could plug in a whole bunch of coordinate points. Just like we've done on these others, you know, like make a T chart, X, Y values, blah, blah, and plug in a bunch of values. But I want you to think about the behavior of these fractions. If we get the cosine right here, let's say we have the cosine X equals a one. See this, so let's say this whole thing is equal to a one. Then we'd have one over one. So secant X would equal one. So right here, that point would also be right there. Now, I'm going to show you what the graph does, and then let me see if you can kind of get an idea of why. The graph is going to go up like that. Almost like it's a parabola, but you have vertical asymptote. So it's not quite a parabola. Kind of looks like it. But it's going almost straight up right here because of the vertical asymptotes. So why is it doing that? Well, get that back there. The reason is let me erase this. The reason is because if you think about how fractions work, if we get a really small number here and we say that this is going to be one divided by a small number, then you're going to have a really big number. So as we get closer and closer and closer to zero and cosine is getting smaller and smaller, then the entire fraction here is getting larger and larger. So that's why this graph is going to be shooting up like this. And so then when we go down here, we're going to have a value of negative one and it's going off towards the vertical asymptotes. So then we can fill in these here as well. This is the secant graph. It has some properties if you look at the cosine graph, then you can kind of be able to tell. So I'm going to put this up here on the cosine graph, and you can see something. Let me switch my colors. So basically, any time you have these peaks, you would have your graph is going to be going up like this. And then you'd have your vertical asymptotes wherever the zeros are. And then the valleys, it goes down. That is your secant graph. Now we're not going to practice how to do the phase shift and what we'd have if we had a number in front of a or shifting it up and down, because it's the same thing we've been doing in the past. But I will say, and if you want to write this down somewhere, that this has the same period as the cosine graph. And so the cosine graph has a period of, so maybe on the side or underneath your graph, that the period is two pi over B same thing as the cosine graph. All right, next one, cosecant. We're going to do exactly the same thing, and that is find the zero. So I'm going to label here on my sign graph, the zeros, because the zeros are going to help me graph the vertical asymptotes, since I'm looking at when does this thing equal zero? Because cosecant is the reciprocal of sine. All right, so now I have my vertical asymptote right here at X equals zero. And where's another one at pi? Oops. There we go. And two pi, and at negative pi, so you can see I'm just putting the vertical asymptotes every place I have a zero. Because that's when the fraction would be a zero on bottom. And now we will do something similar to what we did with the secant graph. And that is wherever you have a maximum. Let me switch my colors a maximum or a minimum. Here, this is what our graph is doing. It will approach those vertical asymptotes. And then down here, where are we? Three pi. Oh, I've got to be at the one value. Like that. And here, and then last one, that was at three pi negative three pi over two, so what I'm doing is I'm looking at the mountain peak and the minimum, the valley is in the mountains. Max and men's to help me figure out where these graphs are. So it's very closely related to their reciprocal sign graph to help you sketch them. Okay, last one, oh wait, go back to this one. Remember that the period for this one is also two pi over B because when we get to the cotangent graph, which relates to the tangent graph, this one has a period of pi over B, not two pi over B if you remember the tangent graph. Okay, so now what happens is let's label our zeros. We have a zero here and negative two pi at zero at pi and at positive two pi. So now this is where we have our vertical asymptotes. So you sketch those again, oh, that was horrible. And this one, and where's another one? Negative pi, and it negative two pi. So we know that as the graph is going to approach a vertical asymptote. Let's do this one here at zero. If we're approaching the vertical asymptote and going this direction, then I know it's either going to go down this way, or it will go up to positive infinity. So which one is it doing? Here you have on our normal tangent graph. If we are left of zero right here, it is negative. So what I'm going to look at is here let me change colors. What I'm looking at is if it's negative, we're getting closer and closer to zero, but it is negative. And then over here, it's positive. So that means as we're getting closer and closer, this direction, it is going down. Like this, and here it is going up like that, because it's positive numbers here, negative numbers here. What about where is it crossing the X axis? That's going to happen. When is this thing going to equal zero? Well, from this graph, if you just think one divided by a number, you're never really going to get to zero, although it will approach zero. So the idea is actually it's where the old vertical asymptotes used to be. But that can be kind of confusing. This thing is going off to infinity. So one divided by infinity basically is a zero, so it's where these vertical asymptotes are. That is where we would put our zero. But let me think about this just a little bit differently. And that is on our unit circle. Let me get rid of the pi over B remember that's the period there. On our unit circle, if we're looking at cotangent, that means it's the reciprocal of tangent. So we're going to do cosine over sine. And so when does that equal zero? This will only equal zero if cosine is equal to zero. If the numerator is, so where is cosine equal to zero on the unit circle, that happens here, and here. So what happens at pi over two, and it happens at three pi over two. And so that's why on our graph, it's going to equal zero at pi over two and three pi over two. And then we could go backwards here and here. Which happens to be exactly the same place where we had our vertical asymptotes up here on this other graph. So now you have the tangent or the cotangent graph should be going down something like this. There is your cotangent graph. So you can reconstruct it yourself if you note the tangent graph looks like, or if you can think through the unit circle, again, we are not dealing with multiplying it by a coefficient in front or shifting it left and right, or a pi, you'll have to do that on the practice, but on this lesson, we've already done so much of that practice. I just wanted to give you the general shapes of the parent functions for this graph. And look, that's it. Pretty quick lesson. Rock that mastery check, and I'll see you back in the next lesson.