 Page 305, I think it was around 304 303 any of these that you were wondering about Yo, Vlad number two Okay, sorry. Yep. So here's tangent Now here's how I know that it's tangent by the way, we're gonna talk about tangent. What do I have memory screen still frozen? How about I do that? Thank you Here's what I know about tangent This is what I have in my brain. I know and I kind of can remember this part tangent goes through zero zero I think that's not too tough to remember Okay, and you could derive that by knowing that tangent is y over x and Okay, I guess You can derive that by knowing that tangent is y over x and when is y over x zero and the top y is zero When is that when you're zero high, which is right here? I Also know that tangent of 90 is undefined. That's in degrees Tangent of pi by 2 is undefined and they get a pi by 2 because I know it's symmetrical in other words. I Remember that and I remember that and let me erase that because they want me to graph cotangent on here. I Also remember the shape of tangent Okay, it says graph cotangent cotangent is the reciprocal So I'm going to graph cotangent first thing is this Anywhere zero high is going to have an asymptote right there right there Right there. How else did we graph? reciprocals was our second step Invariant points hot. What are the invariant points for any reciprocal function? Any wire one and negative one high here's one high. That's gonna stay. That's gonna stay That's gonna stay Here's negative one high. That's gonna stay. That's gonna stay and that's gonna stay Which by the way answers part C when you list the invariant points it looks like It looks like each square is worth pi by 8 if I count because eight squares make up pi Is that correct if I count so it looks like you have an invariant point for example, Vlad at negative 6 pi by 8 comma 1 it's negative 3 pi by 4 in those terms positive Let me check the answers here Well negative 5 pi by 4 Where was I mr. Duke? Yeah, huh? Yeah, reducing the fractions. Okay? And oh in the graph itself Let's see the asymptote would become a zero the asymptote would become a zero The I did this wrong mr. Duke The asymptote of the original would become a zero it's gonna look like this as a matter of fact cotangent Looks like tangent if it's been horizontally reflected and slid sideways That's what cotangent looks like Tell you what I will ask I'm not gonna ask you that Gonna ask you this Or gonna ask you this So what are the equations of the asymptotes if I didn't have the graph in front of me Vlad Here's what I would say cotangent will have asymptotes where Tangent is zero Where is tangent zero? I would often the margins somewhere say well, let's see I know that tangent is zero right there at zero. I Know that tangent has a period of pi so it's gonna be zero at pi Zero at two pi zero at three pi zero at four pi You know we're cotangents gonna have asymptotes then at x equals zero plus multiples of pi Where n is an integer although what's zero plus anything blood? Okay, they probably wouldn't put the zero plus in front. They'd say that's where the asymptotes are. Oh What's the domain of tangent so the domain is what x can't be and I told you it doesn't exist at 90 degrees which is pi by two and what was its period? We have those gaps every pi and This time because that's not a zero. I would leave that in front as opposed to here with a zero vanish What about cotangents domain well cotangent won't exist wherever it has asymptotes X can't be pi n Where n is an integer That's how I'd handle it That's really what I'm gonna ask you about secant and cosecant and cotangent and tangent. I'm gonna ask you equations of asymptotes and Equate and not equation and domains Maybe ranges for tangent there and cotangent the range is all reels I might feel okay asking you for a range of secant and cosecant because they were strange It was two separate ranges we listed last day any others No others Nothing, okay You can write this down a piece of paper somewhere or inside your workbook towards the back or something like this Less than 10.5. Hey, what's on the test mr. Duke? Okay, and your first hint. I would give you is look at the quizzes I've given and your second hint that I would give you is Look at both review Assignments that I Ended out Okay, apparently I can't quite fit that in there we go. Good. No So what do you need to know? You need to know coterminal angles principal angles coterminal angles and principal angles so for example if I tell you that Theta equals 22 pi Over 12 Better yet make it 22 pi over 6 Find me a coterminal angle. How do I find coterminal angles? How do we do it in degrees? Organized mr. Duke by the way. I'm just improvising these notes. They're not gonna be quite as organized But I'm remembering what's on the test. How do I find a coterminal angle in degrees? At 360 how do I find a coterminal angle in radians? At 2 pi what first of all? I think I would take this and I would recognize that this is actually 11 pi by 3 I'd probably reduce it why not get my smaller numbers Okay, and I would take that 11 pi by 3 now Alex You said add or subtract 2 pi except really what would I do if I was clever instead of 2 pi what would I write? 6 pi by 3 why not cut my thinking down? Yes So one coterminal angle is 17 pi by 3 that's the same angle Sorry, that's the same terminal arm not the same angle. You've gone around one more time Principal angles though those were the one between 0 and 360 when we were talking about Degrees 0 and 2 pi we were talking about radians the angle that they gave me is that between 0 and 2 pi? Well 2 pi is 6 pi by 3. So is this between 0 and 6 pi by 3? No, it's too big. Let's find the principal angle Take the angle that they gave you and since it's too big Subtract 2 pi and keep doing that until you end up between 0 and 2 pi My first attempt I get 5 pi by 3 is that between 0 and 2 pi? Yeah, that's the principal angle Remember negative as we go in the opposite direction. I would happily give you a negative one You need to know that you absolutely need to know cast and X and Y and R cast rule X Y and R. I Think enough said we've been flogging that one But you remember I said to you the first four lessons for the remainder of the trig unit All we were really going to be doing is using those first four lessons cleverly I would argue I haven't really taught you anything new for about five lessons. It's just been hey Here's another way you can use X and Y and R Special triangles exact values and we had two main types of questions here one question went like this give the exact value of secant of oh heck 4 pi by 3 This will be a multiple choice question. Almost certainly. Do you know the angle Ellen or are they asking you to find the angle? You know the angle so we can sketch the angle right away We started both of these types of questions the same way the second type of question was went something like this be solved and I would say secant of theta equals over root 3 Zero less than or equal to theta less than 2 pi. I've got these side by side. I hope you can see the key difference We're going to use similar approaches special training Here we would say Well, what's my denominator Emily? So I'm not going to call this pie. I'll call this 3 pi by 3 and 6 pi by 3 And I think 4 pi by 3 there's 3 pi by 3 4 pi by 3 is right there because 5 pi by 3 is right there 6 pi by 3 gets me all the way around. I would write the cast rule I would say to myself self-sandaly secant goes with which trig function Cosine is cosine positive or negative in that quadrant there So this answer is going to be negative I would cross out any positive answers because there's gonna be multiple choice and I would write negative I would find the reference angle. What is the reference angle here sandaly? Pi by 3 and then I would say to myself do I have a triangle with a pi by 3 in it? Yes, I do It's the 1 2 root 3 triangle Sandaly which angle is pi by 3 the bottom one or the top one? right there You said secant goes with cosine Cosine is Hypotenuse over adjacent from good old so katoa days so secant is sorry cosine is Adjacent over hypotenuse from good old so katoa days secant is going to be hypotenuse over adjacent what over what here kiddo? Negative to what I read the over one probably not So I'll give you some question like that heck. I'll even tell you what the angles are going to be Pi by 6 Pi by 3 5 pi by 6 7 pi by 6 11 pi by 6 4 pi it's going to be any multiple of a third or a sixth with a pie in front of it or A fourth because we also had that second special triangle the 1 1 root 2 don't forget that one What did we do here? Well here we started out kind of similar We did a sketch with the cast rule, but here because we didn't know the angle We could only narrow it down to two possibilities Brett secant goes with which trig function and Cosine according to this question was positive or negative Here and here we wanted to find the reference angle here We could just figure it out by doing a bit of subtraction Here we're gonna have to say well I have a 2 and a root 3 do I have a triangle with a 2 and a root 3 in it. Yeah one two root 3 Which of these two angles has a secant of 2 over root 3 the bottom one or the top one? Secant is goes with cosign secant is hypotenuse over adjacent Which of these angles has a hypotenuse over adjacent of 2 over root 3 Justin the bottom one of the top one How big is that angle right there? By by 6 which means this angle is pi by 6 and this angle is pi by 6 Justin. Can you tell me what the roots are? By by 6 that's this first one and are we good enough now that we can just go like that What about how big? Because now you're just seeing it to 12 pi by 6 that don't come common denominator trick very nice If I wrote 12 pi by 6 because I was thinking 12 pi by 6. Let's fix that make it 11 pi by 6 Okay, hey, hey, hey, hey, hey, hey, hey, hey, hey, hey, hey, hey, how about what if I gave you the same question or? Solve secant theta equals 2 over root 3, but instead I said this Negative 2 pi less than or equal to theta less than or equal to zero Zero mr. Do it What if I said instead I want you to find the angles that exist there not the ones between 2 pi and I like this question I like this question Trevor easy Solve it this same way and When you get these look at the first answer and you would say is it between negative 2 pi and zero? Then is it too big? Yeah, it's too big. It's positive subtract 2 pi and Tell you're in the domain that you want so I would say theta 1 is pi by 6 minus 12 pi by 6 and I would say one of your answers is negative 11 pi by 6 is that in the domain that they gave us Yeah, okay What about my second theta is this between negative 2 pi and zero? Say no me too with authority So you know what pick it up and subtract 2 pi and This gives us negative pi by 6 is that in the domain that we want between negative 2 pi and zero. Yeah What if it wasn't keep going? If I give you a really weird domain, I think I'll give you a really weird domain But if they did I just keep going until I end up in their domain Or if I was if they give me a positive really big domain and these were too small keep adding 2 pi The only weird one is tangent tangent Why? Because tangent the period is not 2 pi. What's the period of tangent pi? Say what am I on two special triangles exact values, so some mixing and matching Probably you can expect a quadratic trig with a square and save the square root plus or minus four answers the arc length equation It's gonna be one probably multiple choice How can I remember the arc length equation? What's mr. Do it's brilliant. Yeah, barely nearly godlike contribution to the teaching of mathematics He says although he's never seen this in the textbook the arc length equation looks like which word how convenient not said Alarm Bell number one there was actually two little alarm bells in this unit Prepare yourself Emily Alarm Bell one was when you had sine theta Coase theta secant theta Coase secant theta Equals zeros plus or minus ones Or undefined whoops clean it later Now The undefined is a bit of a fib carrot sine and cosine are never undefined because they had a domain of all reels But these were things that I noticed as my answer. So for example if they said something. I don't know like this solve secant theta Is undefined zero less than or equal to theta less than two five Where is secant undefined? Well, what does secant go with? Cosine okay Where would secant be undefined when cosine was what what gave us asymptotes Ellen? Okay, we're cos theta Equals zero Now we had several ways we could do this One of this one way was to use the human unit circle to recognize that secant was in terms of x and y and r r over x and Then you could sketch that unit circle and you could say when is the x coordinate zero because Maria That's what gives you a fraction with that's undefined when the denominator is zero There and there and you could get pi by two and three pi by two the second way We now have at our disposal is we could actually just sketch one wave of cosine and we could say you know what? secant's gonna be undefined right there and right there Because secant's gonna be undefined when cosine is zero high, which I'm pretty sure if that's pi. This is pi by two and Three pi by two It's y coordinate is negative one boards. It's x coordinate. How far left right is that point right there? That's zero right Gonna be one of these on your written section Something equals either a plus or minus one or an undefined Some where where your solutions are one of these four or more than one of these four corners In fact the written section is gonna look as follows Two graphs, maybe one graph two graphs You know what probably to probably one sign and probably one cosine That's a bit of a tip probably one positive or negative sign and probably one positive or negative cosine what'll be positive what'll be negative okay, and Then after that it'll be two or three exact value trig equations to solve like this one quadratic One undefined and one domain change Yep, you had pi Right, that's how we found the coterminal angle to be added the period What you'll find though Alex is when you start adding pi Often the first time you add pi to your first answer you just get your second answer that you already had and Then you add pi again. Oh now. I've got a new answer and you add pi again Oh now I've got a new answer, but you're gonna find with tangent the two answers that you find normally are already Pi apart exactly and so if you start with the first one you'll find you just regenerate the second one, which okay big deal Already got it. Oh, let me roll in Last thing is what am I on four five six Sketching trig graphs and answering questions about trig graphs, and this is what I want to talk about for about 15 minutes And then I'm gonna shut up and give you a take-home place suppose they said something like this given y equals cosecant of Three theta Tell me the equations of the asymptotes and the domain These will be multiple choice questions, and I won't ask you for both At the same time, but they're both very very related. Here's how I would handle this I Wouldn't I Don't really know what cosecant looks. I know it looks like these weird you shapes going up and going down and remember that But I know where cosecant came from Kara. What is cosecant go with? Fine. Here's what I would say to myself. You know what change colors. Let's do it. I would sketch Which Kara, I hope by now you can almost do in your sleep. Hopefully you know that Where will cosecant have asymptotes Where sine is how high? Sandley So normally before the three the answer here would be x equals zero plus multiples of pi Where n is an integer although they probably wouldn't write the zero plus in front and the domain would be x can't be zero plus multiples of pi where n is an integer except they probably wouldn't write the zero plus in front because zero plus pi n is just pi n Now that would be if I just had cosecant of theta what do I have here? I don't just have a theta Trevor. What do I have here three theta? Horizontal or vertical? Because it's next to I got to be careful. It's next to the x I was gonna next to the theta, but you know it's next to the where the x goes. I think I would say this. This is a horizontal Compression by a third Compress them by a third. That's the same as dividing by three. So this is your final answer. This is the Equations of the asymptotes or the domain of cosecant. secant Little trickier given y equals the secant of oh heck x over 2 Tell me the asymptotes Tell me domain. The short answer here is I don't know. I derive it I know that secant goes with which trig function Tyler So you know what I would sketch cosine Which hopefully you know by now looks like that Where that's 2 pi That's pi and this is Pi by 2 and 3 pi by 2 and I would say secant is going to have asymptotes where cosine is zero high Normally secant would have an asymptote x equals pi by 2 Plus multiples of pi it would have a domain of x can't be Pi by 2 plus multiples of pi where n is an integer. Is that okay so far Brett? Okay You want to give the young fella next to you a little elbow. I think he needs it. Oh, he's with me. I Don't know you were over there somewhere gazing off at this base Joel. I don't have an x here though. What do I have here? x over 2 Horizontal or vertical next to the x what is dividing by 2 everything's backwards. What does that actually do a horizontal? What? This is actually Horizontal expansion by 2 So the actual answer is going to be multiply that by 2 Multiply that by 2 or if I ask for the domain, that's what x can't be Multiply that by 2 Multiply that by 2 That works for tangent or cotangent or so if they give me cotangent Brianna. I think about tangent I said you need to know the asymptotes are at positive negative 90 degrees plus or minus pi by 2 My domain is x can't be those and then I just do whatever transformation I use the unit one step rather than try and memorize a whole new method So that's pretty much most of what's going to be on your test I have one more thing we need to do before I completely turn you loose and Bear with me for one second here. Can you get out trig review number one, please? And what I'm gonna do when I gave you some questions to try when I was away I skipped all the asymptote reciprocal trig function questions I'm just going to tell you which ones you can also try for good practice because the textbook unfortunately Doesn't have a lot of good ones. So I Had put this list up The other day when I was away I said questions that are fair game and I'm just going to add here Also, and you can either circle these or write these down Nope Bear with me. I already signed 11. Yeah 12 Bear with me like there were some I did assign 49. Yeah, I did good 58 ugly good 75 Really? Yeah, as a nasty 90 Which I don't think I signed last time. No, I did not did I sign 92? Nope good 92 We're missing a bunch of questions on this for some reason almost looks like certain pages didn't photocopy or couldn't print I didn't catch that until now. Oh well So with that one you can try those and then if you want to get out trig review number two also Two This is trig review number two. By the way for some dumb reason. I said you could do number five But you cannot do number five. Let's go up here You can't do number five yet. That's a trig identity, which is next unit. That was silly 1314 I already said sure 22 27. Oh hang on. Mr. Dude. Don't type there. You're typing here 22 27 So you can add those four Okay, last I have a take home quiz for you. Let me grab it and I hand it out