 Lesson 10. What we're going to look at today, Kirsten, is tangent a little bit, but really cotangent, cosecant, and secant. Really secant and cosecant. We're not going to spend as much time on tan. Then what I'm going to do next class is really let you know what I expect you to know about tangent and cotangent, and secant and cosecant. Probably won't really expect you to graph them. I might expect you to recognize one of the graphs, but I don't think I'll ask you to actually graph it. Well, secant and cosecant, cady and cotangent are reciprocal graphs. So it would be hubus. That means it would be wise to review what we remember about reciprocals. So you're ready? Reciprocals, when my original graph was zero, my reciprocal had a vertical asymptote. When my original graph was zero, my reciprocal had a vertical asymptote. What's this graph of? Sine. So here's y equals sine x. What's the reciprocal of sine? Which reciprocal trig function goes with sine, secant, cosecant, or cotangent? I'll do this in red. I already know that wherever sine has a zero is zero high, it's going to be asymptotes. We're going to try and we're going to graph this later on in our notes, but let's try and anticipate what it's going to look like. What else did we say? Well, when my original graph is positive, my reciprocal is also plus ve positive. In other words, if your original is above zero, the y-axis, your reciprocal is still above the y-axis. Don't suddenly go below the y-axis. Oh, and when its original was negative, my reciprocal was also negative minus ve is my abbreviation for negative. In other words, over here, I'm pretty sure there's going to be some red graph down here and some green graph here, but I don't think there's going to be any green graph here. I don't think there's going to be any red graph here. I don't think there's going to be any red graph here. There's no reciprocal there and there. Ah, then we said this. When my original was one high, my reciprocal was one high, right? When my original was negative one high, how high was my reciprocal? Also, So negative one. We said these were the fancy word, remember this word, Tyson, invariant points. We said anywhere plus or minus one high, which is really important because Amanda, what's the amplitude of sine and cosine? So you know what? There is always going to be a key point or two plus or minus one high that's invariant. Oh, by the way, that's pi, pi by two, three pi by two. Then we said, imagine you're a little bug standing on the graph. When your original gets bigger, the reciprocal gets smaller, decreases. When your original gets smaller, your reciprocal gets bigger. Oh, and then remember, remember, remember, remember, Nicole, when my graph got really small, got close to zero, my reciprocal got really big. It shot up to either positive or negative infinity, depending on whether you're getting closer to zero from below or closer to zero from above the x-axis, right? And the graph approaches a vertical asymptote. Now we have to add a tiny bit more because when we did transformations in unit one, I did not give you graphs with asymptotes on them. I gave you graphs that were fun that may have had asymptotes when you did the reciprocal, but I never gave you an asymptote and told you to take the reciprocal. But tangent, your cousin, does have our asymptotes already. Where tangent has asymptotes, where it shoots off to infinity, my original graph is going to get closer and closer and closer and closer to zero. So we're going to say if you have an asymptote already, you're going to approach zero. And in fact, you're going to start to level off just about. You're almost going to get very briefly a horizontal asymptote. You're not going to approach zero like you're going to kind of and hit zero and then you'll see in a second. By the way, we have to reemphasize. This means inverse doesn't mean reciprocal. Inverse is when you wanted to find the angle if you were given the decimal value or the radian value and you had to go shift side or shift closer, shift hand to find the angle. How are we going to sketch the reciprocal functions? Something like this. Watch. Let's see. Remember my little invisible bug? As I move to my left, Ryan, am I getting closer to zero or further away from zero as I move to the left, the green graph here? The green graph? I think he's getting, isn't it getting closer to zero? Hi. So my reciprocal is going to do what? Get further. Noink. How about as I move to the right, Ryan, my green graph, my original is getting closer to zero. What's my reciprocal going to do? Yoink. Oh, and you know what? Yoink. Yoink. That's what cosecant looks like. Cosecant looks like that. Honestly, I hardly have it memorized. You know what I know how to do, Matthias? A reciprocal and I can graph sine and cos in my sleep. So if they ever asked me to do secant or cosecant, I don't. I quickly sketch the sine or the cos sign that goes with it and anywhere one high stays invariant. Bigger becomes smaller, smaller becomes, and I do the reciprocal. So here's sine x and we've graphed it between negative two pi and positive two pi. It says graph the cosecant. Okay, first thing we would do is we would say anywhere zero high there's going to be an asymptote. We're on page 302. Home, we're here. Sorry, go get it. Invariant points, anywhere how high? One or, careful. Well, let's see. That's not going anywhere. That's not going anywhere. That's not going anywhere. That's not going anywhere. Bug. Getting closer to zero, yoink further from zero, shoot off to infinity. Closer to zero, yoink further from zero, shoot off to infinity. Closer to zero but negative, shoot off to infinity but negative. Closer to zero but negative, how about we shoot off to infinity, but negative. And then it repeats, looks like that. Hey, let's graph it on our graphing calculator, it's just a C. So, clear whatever graphs you got and make sure you're in radians. How would I graph cosecant? I don't have a cosecant button. How do I type in cosecant? No, not second function sign. All right, I gotta freak out on you. I gotta freak out on you and say, no, that does not mean that. That does not mean that. That does not mean don't do that. Thank you for letting me freak. I missed that. How do I graph? How do I graph cosecant? Oh, one divided by sign X. But don't hit graph yet. Let's change our view windows to match theirs. So once you've typed that in, hit window. How far left does their graph paper go? Oh, let's do that as our X min. How far right does their graph paper go? Two pi, we say. Let's type that as our X max. What's their scale? Count, figure it out. What's one square worth? Figure it out. The easiest way to figure it out, Nicole, is to count how many squares make up pi. How many squares make up pi? So what's one square worth? Oh, let's do that as well. How low does this graph go, Nicole? What's their scale? Actually, technically sort of one third, which is a yucky, is three squares make up one. You know what? Vertically, I'm never as fussy because I'm not that... Anyways, hit graph. What do you get? Do you get the same as me? It depends. Who's got the 84s? Who's got a TI 84? Anyone in this class? Do you get the same exactly as me? So you don't have that, is that right? Or is that a different issue? I have no idea what you said. One line missing. You don't have that one or that one? Look out. Do you want to go under that one? So yours has better software. Anybody else not get this at all, first of all? Let me come help you. So we've graphed it. We've double checked on our graphing calculator a little bit. Looked like that. What kind of questions might I ask you? One thing I might ask you is to tell me the equations of the asymptotes. Now there is an asymptote. There is an asymptote. There is an asymptote. There is an asymptote. There is an asymptote. What are the equations? Well I can't list them all because there's an infinite number because this graph keeps going forever and ever and ever. Instead I'm going to list one and then spot the pattern. First question. What is the equation of a vertical line? Math then. It's equation of vertical line and I also did it at the beginning of this year. No one knows, really. You need to know this. Okay. One, two, three. That's a vertical line that goes through right at x equals three. What's the equation? What? You mean when I say that's a vertical line that goes through right at x equals three I'm actually saying the equation? What's the equation folks? Cassandra, what's the equation? Ready Cassandra? Here we come now. Here we come now. What's the equation of this line right here? That's not an equation. It's not what you said to me. That's a number. What's the equation? Loud voice please. That's what I'm looking for. There's the first one, the most obvious one. X equals zero. Where's the next one Cassandra? Then after that are they pi apart? X equals zero plus multiples of pi. And then we have to add though where the number n is an integer to say, look don't multiply pi by 1.5. It's not every 1.5 pi. It's every one pi, two pi, three pi, four pi, five pi, six pi, negative one pi, negative two pi, negative three pi. Oh, and then there's one more thing. What's zero plus five? Five, what's zero plus six? What's zero plus pi n? They probably almost certainly won't write that in front. But I just wanted to show you how we got that. And if you're not starting at zero, phase shift, you might have to start at plusing with something. What are the invariant points? Anywhere plus or minus one high. By the way, really it's X equals pi by two and every two pi in either direction. Sorry, every pi and no, every two pi, 12 squares. Yeah, two pi and X equals, I don't know, three pi by two and every two pi in either direction. I'm not gonna worry about any invariant points. I'm not gonna really be fussy on that to ask you then. We're useful to graph though. What I will probably ask you is domain and range. First of all, for sine, what was the domain of sine? All reels, right? What was the range of sine? Remember it was nice, negative one, less than or equal to Y, less than or equal to one between negative one and positive one. Sadly, cosy can't, not so nice. Domain. Well, it's sort of all reels, except there's gaps. Like it does go forever. Where are the gaps where the asymptotes are? Where is the first asymptote, Cassandra? Then, then, okay, so the domain, now the asymptote, you're saying what's the actual equation of those vertical lines? The domain, you're saying X can't be. X can't be pi n, where n is an integer. That's your domain. What's your range? Well, your range, okay. Andrew, how high does this graph go? How low? But there's a gap, see it? In fact, the only way that you can do this easily, there's a yucky way to do this in one line, but the easiest way is to say, you know what? I'm gonna do the range of this one. Everything above and touching how high. Everything above and touching one, comma. And then I'm gonna do the range of this one. Shannon, everything below and touching how high. Don't say one, careful. Below, less than, touching. There is a way to write that in one combined yucky statement, but I'm not gonna do it that way. It's easier just to say, you know what? It's kind of two separate graphs with a gap between them. That's cosecant. What would secant look like? Good question. There's cosine. I guess secant would have an asymptote right there, an asymptote right there, because that's where cosine is zero high, right, Tyson? It would be invariant right there. It would be invariant right there. It would be invariant right there, because those are plus or minus one high, right? And getting smaller, you know, getting bigger, getting closer to zero, shoots off to negative infinity. In fact, it looks an awful lot, like this guy slid sideways, because this guy has its first vertical, its first invariant point at pi by two. This one has its first invariant point at zero, or it's over here depending on how you wanna think about it. They look similar. What if we muck around with the, well with this? Oh, let's add one more thing. What's the period of cosecant? Two pi. That hasn't changed. Amanda, what's the period of cosecant? Woo-hoo. What about here? What about here? What if we replace the x-width in this equation? Two x. Do you remember what that did to a graph when you replaced an x with a two x? First of all, horizontal or vertical? Horizontal, okay, okay, okay. Horizontal what? Let's see how that affects things. What was the domain of cosecant? Well, we just have it on the previous line. It was x can't be pi n, where n is an integer. You know what the domain of cosecant of two x is? It's, so read that to me. X can't be pi n over two, where n is an integer. What if they put a three there? That'd be a horizontal compression divide by three. What if they put a one half there? That'd be an expansion by two, times by two. That give us a, you can memorize this, but I think we can actually tie this into something we kind of know, and it's good review, Katie, of some of the unit one stuff from way back when. Katie, this word right here, read it out to me. Will that change a range ever? Because I think range is vertical. Okay, you know what? Range is still the same. Y greater than, or negative, not negative one, that's positive one. Everything above and touching one, comma. Everything below and touching negative one. Everything above and touching one, comma. Everything below and touching negative one. What if we say the period of cosecant was? Amanda, Amanda, joy of my heart, light of my love. Read this to me please, light of my love? Oh well, what do you think? Because period is a horizontal. Yes, yes, yes, yes, yes. You're right, what? All right, two pi over two, so you can see the math that you did in your, yeah, pi. All right, Amanda, let's expand that. Hey, what if there was a five right there? What would the period be? What if there was a one half? Now that would be an expansion by four pi, yeah. Okay? Oh, could you even holly handle something algebraic and just think about the math that we did of it? I put like a A there, it'd be two pi over A. Oh, or two pi over B, if I wanna tie it into what we've already memorized, the period as being two pi over B, that's really what it is. What about the asymptotes? Well, the original equations were the same as the domain except it was, that's where they equal pi n where n is an integer, not equal to pi n over two where n is an integer. What we did here is we said, you know what? The graph of cosecant of two x is a transformation of cosecant by a horizontal compression by a factor of, and you know what? I wrote by a factor of two up here. I should have said by a factor of a half, shouldn't I? Because that was our notation by a half which was the same as timesing by a half which was the same as dividing by two. And it is a compression about the y-axis but I never asked you to remember that I just, because that confused me. In fact, compared to the asymptotes of cosecant, cosecant two x, the asymptotes are twice as frequent. They happen way more often. I'll show you. One divided by sine of two x because that's cosecant of two x. And I'm gonna go back arrow and I'm gonna hit enter to make this line thicker so it stands out. Graph. Yeah, asymptotes are twice as frequent. Period's half as long. Turn the page. The absolute value transformation. Remember that one? When we did this, y equals the absolute value of whatever graph we were dealing with. Remember what that was? Because that was a fairly, we did that at the very end of transformations and the reason we said it for the antithesis that was one of the easier ones. Anything that was positive nefanyl remember stayed as is. What happened to anything that was negative? Clipped up. Okay. When my original is positive above the x-axis, the absolute value is identical. Fancy word invariant. Oh, and Sabrina, when my original was negative, the absolute value is a, I think what they want us to write here nefanyl is a vertical reflection. It flips up. I like your explanation better. It became positive, flips up. Okay. Here's cosex. What would this look like? Well, that would stay. That would stay. And that would stay. You know how I know? Because they're positive already and the absolute value of positive is positive invariant. But nefanyl, what would happen to this little guy right here? Right there, yoy. Right there, yoy. Sort of looks like a little ball bouncing but it's not quite because the physics nerd within me knows that a bouncing ball doesn't actually graph an absolute value of a cosex. But you can use that as a memory tool if you want. Stuff to try. Number one, Mr. Dewick, there's tangent. Yeah, I know, ugly cousin. We'll talk about this tomorrow. I haven't talked about a phase shift. This moves pi by four left. Nicole, pi by four what? Will that change the range? Will it change the period? No, because the period length won't change. But you know what it's gonna change? Move your domain left, move your asymptotes left. See if you can figure that out. If not, I'll talk about that next class. So did I circle three? Let's circle three. Let's circle four. Try five, try six, try seven, nine, 10. And we'll pause there, but I also think I have a take home quiz for you. Let me double check.