 And it should begin. Awesome. I'm just going to shut the doors. Hopefully Trig is a little more familiar, a little more ingrained in your memory. Thank you, sir. Little comment, I'm not allowed to use people's names for anything I post online. So today you're all going to be Johnny, if I point to you. I just find I've got to be able to talk, except for my young friend Zanzibar back there who wants to be addressed that way for internet posterity. We're good with it. All right. Trig, what did we start out with? Well, the first thing that we started out with was rotation and reference angles. I'm going to try and flip between typing and handwriting because my handwriting is atrocious. I'm slowly trying to master moving from both worlds, Johnny, to see if I can pull it off. So let's see how this works. We would ask you, for example, stuff like this. Kind, sir, could you hand out the test booklets? Because we're going to be probably doing some of the Trig questions from there, too. I forgot to hand them out. Example, find a reference. You know what? Can't type that one because I don't know how to do a pie in this. For example, stuff like this, example one. Find the reference angle for negative 8 pi by 3. Find a reference angle for that. So we introduced very quickly to you the union of graphing and Trig with that unit circle idea. An awful lot of the time, we would do a little sketch. Now, I always teach my kids to keep my memory, to keep my math as easy as possible. I always found the common denominator, and I'll show you what I mean. Most of my students would have said, OK, that's pi, and that's 2 pi. No, they wouldn't have. I never teach my kids that. I start out that way. But then, Johnny, I say this, what is my denominator, Zanzibar? You know what? Instead of pi, I'll call that 3-thirds pi. Isn't that still pi? And I'll call this 6-thirds. Isn't that still 2 pi? It cuts down on my thinking a lot, because now negative 8 pi, hey, there's negative 3. There's negative 6. How much further do I have to go? 2-thirds, negative 2-thirds. You know what? It's going to be right about there, ish. Now, I've got the quadrant. What quadrant am I in? Quadrant 3, what's my reference angle? Remember, reference angles were always measured to the positive x-axis. So my reference angle is going to be positive. Yeah, pi over 3. That's a good multiple-choice question. Can you see OK? That's a good multiple-choice kind of a question. We would have probably pi over 3, negative pi over 3, 2 pi over 3, and negative 2 pi over 3. Those would be your four answers to pick from, probably something like that. We also talked about coterminal angles. Can you give me the smallest positive coterminal angle? So example 2, what is the smallest positive coterminal angle for the above question? Doesn't like the word coterminal? Fine. Now you're happy? No, apparently not. Doesn't know math. Oh, well, coterminal. I'm calling it that. What were coterminal angles? Those were angles that had the same terminal arm. What is the smallest positive coterminal angle? That one. How big exactly? 4 pi by 3. Do you remember the fancy word they used for the smallest positive coterminal angle? It's also the person in charge of a school. The principal angle spelled differently. But the principal angle is the smallest positive coterminal angle. So that was coterminal angles, reference angles. We also did a quick review of sine, cosine, and tangent. And we introduced to you the reciprocal trig functions, secant, cosecant, and cotangent. Tangent goes with cotangent. That's easy to remember. What does cosecant go with? Sine. What was the way to remember? Cos never went together. Then we moved on to what we called the cast rule. And let's look at a few questions. If you could open your little test booklets to page 20, please, page 20. And the first one we're going to look at is, I can close this, number nine. Too far. So it's probably page 21. You got a little booklet? Page 20? You can share, right? Don't write on these. Don't write on these. Don't write on these, please, because I don't have extras. All right. In which quadrant is the terminal arm of an angle 8 pi by 5 radians? Well, the nice thing is, the answer is either 1, 2, 3, or 4. What would you do? Quick sketch. Oh, wait a minute. I can cheat and use my technology. Lovely. Johnny, what's my denominator? So I'm going to call this 5 pi by 5, 10 pi by 5, 8 pi by 5. Is it here or is it here? How can I figure that? Well, split the difference. How far between here and here, Johnny? How far, how many fives between here and here? How many pi by fives? If I kind of eyeball 1, 2, 3, 4, 5, I think that's roughly dividing it into fifths in my head. I'm pretty sure it looks like 1, 2, 3. I think 8 is 3 fifths further. Sorry? Question? We're good. I'm pretty sure it's in quadrant 4. By the way, what can you tell me about quadrant 4? Which trig function or functions is positive in quadrant 4? Cosine and? What does cosine go with? Secant, right? Don't forget that. Let's be pretty clear. The odds are pretty good. They're going to give you a reciprocal trig function as the hint in terms of what quadrant you're in. They're probably not going to give you, oh, where is cosine and positive? Too easy. That's just writing down the cast rule and regurgitating the answer. They're going to say, hey, where is secant positive? Or where is cosecant negative? Or something to that effect. Let's look at number 10 real quick, turning the page. So which pair of angles is coterminal with that bad boy? You know what I would do. By the way, I don't mind you not writing anything down and printing this up, but you probably might want to have a scrap piece of paper to try somebody's on, right? I'd be sketching this right now. I'll do that. You know what? That's 9 ninths. So 11 ninths looks like it's 2 ninths further. And that would be 18 ninths as well as 0. Let's see. 29 ninths, 18, 27, 28, 29. Ooh. I like that guy. 7 ninths, nope. Negative 7 ninths, nope. Oh, you know what? I'm quitting. Oh, I would have also crossed that one out right away. Anyways, I got the right answer. I usually don't waste my time trying all the possibilities. I'm confident enough, and I know that the provincial exam is a bit of a sprint. I'm confident enough that I'll say, I'm not going to try C and D. I know I'm right. And if you're a C student, maybe you want to try out more of them. If you're an A student, you'll be able to find the right answer pretty quick. Sorry, I'm just getting familiar with some of these. Yuck, yuck, yuck, yuck, yuck. Oh, we can certainly do number 12 with the cast rule. So number 12 says, which of the following is true if secant is negative and tangent is also negative? Now we really do need to write out the cast rule. C, A, S, T. Let's do the easy one first. Where is tangent negative here and here? Yeah. Where is secant negative? No idea. Oh, but what does secant go with? Coast. Oh, this is really saying, where is cosine negative? Where is cosine negative? Here and here. Where's the overlap? What quadrant am I in? Quadrant 2? Now that's not what they're asking. They want to know the domain of what the angle could be. A, B, C, or D? What is it? Zanzibar? B. Very good. So that was the cast rule. Have I got any more on the cast rule really quickly? Let me look. Then we looked at applications. We defined sine, cosine, and tangent, but now we define them in terms of a graph. I think we did it when we did the cast rule. We said that sine theta, in terms of the graph, was what over what? No longer just opposite over hypotenuse. Y over R. Cosine theta was what over what? X over R. And tangent theta was what over what? You've got to have these memorized to the point where they're really on the tip of your tongue. I don't freak out if you don't have secant, cosecant, and cotangent memorized instantly. In fact, I don't have them memorized. I'd still derive them every time. So if somebody said to me, hey, duic, what's cotangent? I don't know. I still go to tangent, and I say, well, tangent is Y over X. Oh, what's cotangent? X over Y. What is secant R over X? And what's cosecant R over Y? We can use that. So what kind of questions do they love to throw at you? Oh, stuff like, well, certainly number 13. We're not going to do it. But it says, determine the ratio for cosine if sine theta equals J over K and cotangent theta equals negative H over J. Well, OK, let's do that one. 13. It was a little more to that one than I thought. What you want to remember is, especially on the multiple choice section, if they ever give you a trig function as a fraction, or as a whole number, because if it's a whole number, what's it automatically over? One, that is a fraction. They're giving you X and Y and R, two of them, always. In this case, sine, sine is what over what? I think they've told me that Y equals J, and that R equals K. Not only that, apparently, Y is positive. Cotangent is what over what? So I think they just told me that X equals H. No, no, no, they told me a little bit more than that. What else do I actually know? Negative H, because that negative had to come from somewhere, and by reasoning, it didn't come from there. Now I can answer the question, which trig function did they want me to find? Cos theta, which is X over R, negative H over K, the answer is B. Nice question. 14, another classic example, I think. 14. Now when I first read 14, no, I'm not going to panic. I'm OK, I can handle this. Says, in standard position, the terminal arm of angle A passes through negative 2K comma negative 5K. Oh, and by the way, K is positive. Find the exact values of cosecant and tangent. You know what? I think what they're actually telling me is X and Y. Tangent I can get right now. Tangent is what over what? Y over X, negative 5K over negative 2K. What does that simplify to? What's a negative divided by a negative? Positive. Can I cancel the Ks? You know what? That's wrong, and that's wrong. I've just turned this into a true and false guessing question, if nothing else, because now I've only got two to pick from. Cosecant is what over what? I don't know. Cosecant goes with which trig function? Sine is what over what? Y over R. Cosecant's going to be R over Y. Got it. I'm going to write that down. Cosecant A is R over Y. And that's what they want me to find. I know Y. What's Y? What's R? Do you remember? Johnny. We said that R squared equals X squared plus Y squared. Although Johnny, I usually got lazy and said, you know what, it's that. Now this looks scary also, because there's no case. I'm not going to panic. Maybe the case cancels. I'm not going to freak out. Let's see. This is going to be the square root of negative 2K squared plus negative 5K squared. What's negative 2 squared? I get 4K squared, because K squared is K squared. Plus what's negative 5, 25, and what's K squared? I get this, square root of 29K squared. I don't see any case. What did they do? I'm not going to panic. What did they do? Sorry. I think they wrote R as the square root of K squared, the square root of 29. Yeah. By the way, what is the square root of K squared? They did math 10. Remember simplifying roots in math 10? Except they did it with variables. OK, no big whoop. It's K root 29, and now I can do the entire trig function. That's R. That's Y. Cosy cant theta, or A, is going to be K root 29 all over negative 5K. K's cancel, because it is factored. And I get negative, because it's positive divided by negative, root 29 over 5. Nice little question. So I get that there. By the way, do you realize I didn't have to do any of that? I could have stopped right here. Here's how come. If this is negative and this is negative, what quadrant do I have to be in if my x-coordinate and my y-coordinate are negative? Sorry, say. Oh, if I'm in that quadrant there, is cosy cant positive or negative in that quadrant there? Negative? Oh, I could have just jumped right to that, because I knew it was one of these two. I could have skipped all that finding R. But I wanted to walk through it with you, because they might not give you one that has that flaw. To me, this is a flawed question. I'd phrase it differently. Did any of you notice that, by the way, that you didn't have to do all that? Cast rule and applications. Um, no, no, no, no, no, no. That's graphing. That's graphing. That comes later. OK. Then we looked at special triangles and exact values. We also looked at radians. I forgot to actually include that as a separate topic, because remember, we did degrees for two lessons, and then we ditched it permanently. So I hardly even remember doing degrees in math 12. It was radians, radians, radians, radians, and I should mention that, because I can tell you all one question you're going to have on your mock exam right now. Oh, and one question you're going to have on your provincial exam right now. In fact, one question you're going to have on every one of your mock exams. It's going to be, there's two multiple choice questions that appear every year. One is in logarithms, and it says write the following as a logarithm, or if I write the following as an exponent, that's on every year. And the other one is go from radians to degrees or degrees to radians. So just to jog your memory, I think, number one, let me go check. Ah, there you go. Number one, express four pi by nine radians in degrees. Shows up every year, often shows up at well. It used to always be number one. Now that they've gone to a non-calculator section, sometimes they'll put this on the non-calc section if it works out really, really, really, really nice. Or sometimes if they want you to go with decimals, you put it on the calculator section. Cliff, is there anybody here who would glance at this and know it's 80 degrees right now? Because it is. My kids in their prime could, let me show you the shortcut now. Here's what I drum into my brain. Pi radians is the same as how many degrees? Pi radians is the same as how many degrees? 180. This is dividing those pi radians into nines. What's one ninth of 180? In other words, what's 18 divided by nine? 20, 180 divided by 20. Do you know how many 20s they want? That's how I got the 80 degrees zzap in my head. You can do that if they have nice denominators. If you notice that this bottom number goes into 18, you can often just do it in your head. Now you don't have to, but if you want to be the uber nerd like me and when you grew up, there's how you can. Otherwise, if you need the conversion factor, here's what we said. You would go four pi by nine. And although normally we don't write units for radians, I'll just write it in this so you can see what we're doing. We're gonna multiply by one by a funny looking one. I want the radians to cancel. I want degrees to end up on top. So I guess here is where I have to put the 180 and here's where I have to put the pi. Can you see, Johnny, it's gonna be four times pi times 180 divided by nine, divided by pi. Ooh, as a matter of fact, for what it's worth, not only do the radians cancel, hey, the pi's cancel. And now you might actually say, oh, 180 divided by nine, 18 divided by nine is two, 180 divided by nine, hey, that's 20. Can you see the 80 coming out at you? Oh, units, degrees. You can also go from degrees to radians. That one you're all pretty good at because you've done that common denominator trick when you're sketching your circle quite often. I don't know if you thought about it, but that's also what I was kind of sneaking in there. Every once in a while, they will go with decimals and you have to use your calculator. For example, they may say something like this. Write 2.67 radians in degrees or 117 degrees into radians. Well, again, you would go 2.67 times. I want the radians to cancel. I want the degrees to be left behind. I guess the 180 has to go on top. The pi has to go down there. The radians cancel. Thanks for coming. So I'll throw this online. Now I would go straight to my calculator. I would go 2.67 times 180 divided by pi. This would be a multiple choice decimal kind of a question. 2.67 times 180 divided by pi. 153 degrees almost bang on. I'd like to think that I was doing that math in my head, but I'm not that good. By the way, what unit are we doing right now? Trig, are any of you in physics 12? Time to put your calculator in radians, right? Let's do that before we forget. Those of you in physics 12 and in math 12, during your mocks and during your actual provincials, please first thing, check to make sure your calculator is in the right mode. I had a student a couple of years ago. I can't use his name online. I'll just say Johnny. And with about 10 minutes left in my physics mock, he was working and all of a sudden you just heard this. It's the sound that I would imagine a dying cow would make as it was being run over by a tractor. Mm-hmm. Literally, we've been going for about an hour and 45 minutes. There was about 15 minutes left. And I just smiled and said, folks, just make sure your calculator is in degrees, not in radians, otherwise you'll be getting all the wrong answers in physics. And I didn't even have to look. I knew exactly what that moan was because it was just this anguished soul going, this is why I've been getting all these weird answers and couldn't find any of my answers on the multiple choice. Oh, I just want to say his name for posterity. My students know who it is, but I can't put it online. Here we go. Exact values. We said there was two special triangles that you had to memorize. So we had two special triangles. I think Mr. Kamosi writes his vertically, I write mine horizontally because I'm a lefty and just, I don't know. By the way, there's technically three triangles, but two of them are the same, so I just memorized two. We had the one, one, root two where each of these angles was how big? 45, that's degrees, that's blasphemy. Get the behind me, sorry. Pi by four, pi by four. This allowed us to find that the sign and the cosine of pi by four were both one over root two and the tangent of pi by four was one over one. Then we had a second one, sorry Mr. Kamosi, but I always draw mine like this and I yelled at my students, I said draw elongated, here's what drives me crazy. Every year I see some kid who draws that triangle like this and then they can't tell which is the smaller angle and which is the bigger angle. If you exaggerate it, first of all it's one, two, root three, but here's why I exaggerate it. It's pretty obvious which is 60 degrees and which is 30 degrees. Which one's bigger, the top one or the bottom one? Now I apologize for swearing, I just said degrees, let's go to radians. 60 degrees is what fraction of 180? You know what, pi by three, pi by six. I marked a question, a written question, a couple of times in a row at the provincials and it was amazing how many kids could do the whole question but they drew a small triangle and labeled it wrong. It just drove me crazy. I was yelling at their teacher at the paper, why are you just showing me stretch it out and it takes care of itself, it's a simple fix. Right, then you'll never get it wrong. I mean unless you can't clue in that those are bigger than each other but then you got other issues. Fingers am I holding up, you don't three, right? Once you had that then, I could ask for the exact values for certain trig functions and that was your trigger phrase. Your trigger phrase was either special triangles or give your answer as an exact value. Now, I fibbed a little bit. That meant your answer was either from a special triangle or your answer was from an angle right there, right there, right there or right there. Your answer was from an angle of pi by two, pi, three pi by two or two pi or the negative variations. These ones we usually got, well you could either get it with the unit circle or by doing a quick sketch of the graph. Anything else we got? Anything else we got from special triangles? For example, give the exact value four, cosecant of five pi by three. What we did is the very first thing and probably you do this instinctively, you're all pretty good, we actually said do I know the angle or am I being asked to find the angle? So being asked to find the angle that was doing everything in reverse, we had to try and figure out what quadrant we were in, try and figure out the reference angle and then work our way to what theta one and theta two were. Here, I'm pretty sure they gave me the angle. I'm just trying to get the trig ratio. So this was a little easier. I still started out by going hey, five pi by three. Well, there's three pi by three. Five pi by three, okay, there's four pi by three. I think five pi by three is probably right about there. Again, I divided that bottom roughly into thirds. What quadrant am I in? Quadrant number four, C, A, S, T. Cosecant goes with which trig function is signed positive or negative in that quadrant. And again, if this is a multiple choice question, you probably just got rid of half your wrong answers and turned it into a true-false question. What's my reference angle? Pi by three, have I memorized a triangle that has a pi by three in it? I have? I would sketch it now. I know I have it right above me, but just in case this is a brand new question, I would quickly go, okay, sketch it, do it, yells at me, says do it nice, elongated one, two, root three. Which angle is pi by three, bottom or top? Top one. Cosecant goes with which trig function? Sine, sine was what over what now in terms of soca-toa. Opposite over hypotenuse, so cosecant is gonna be hypotenuse two over opposite root three. If this was a multiple choice question, Johnny, I can even tell you what four answers you would have. Positive two over root three, negative two over root three, positive one half and negative one half. I'll probably use the pi by six, pi by six, in case they got that one. More than likely. So let's look at a few from this little practice test. What can they throw at you here? I don't know. Oh, well, number two says find the exact value for cosecant to four pi by three. That's really similar to the exact value for cosecant to five pi by three, but I bet you it's just negative, positive or the same. Four pi by three would be in this quadrant here. Is cosecant negative or positive in this quadrant here? You know what, same answer. So I'm gonna unfortunately skip that because I didn't realize, oops, oh, Jim. Yeah. Special triangles, special triangles. Lovely written section. Nah, nah, nah, nah. We're gonna come back to it anyways when we do solving tricky equations. Arc length. Sorry, Johnny. AR theta, unfortunately that's not an equation. That's three letters, not separated by an equal sign anywhere. Yeah, again, this is Mr. Dewick's humble little contribution to the teaching of mathematics in grade 12. Mr. Dewick says that the arc length formula looks like the word arc. If you're looking for a dumb way to remember it, that works for me. Someone else came up with something else, but I can't remember. Sorry, Johnny, what? Okay. Oh, yeah, only one thing you have to, okay. First of all, A stands for arc length, R stands for radius. The only thing you have to remember is theta has to be very, very specific. Has to be in radians. What are the odds of them giving it to you in radians on the provincial, do you think? Pretty slim. They're gonna give it to you in degrees just because it gives them more wrong answers to make up on the multiple choice section. That's a terrible way to be testing, but sometimes you just gotta do what you gotta do. So, I think, number 11, yeah, there we go. 150 centimeter pendulum swings through an arc which measures 100 pi centimeters in exact radians through what angle does it swing. Now, if they really, really want it to be nasty as an extra step, they might carefully pick an answer and then they might say, give the exact sine value of that if they made sure that one of your answers was one of your special triangle angles or trying to think of weird ways that they could extend the arc length formula. Let's do this one first of all. I would say, hey, Newick says it looks like the word arc. What are they asking me to find? The arc length, the radius, or the angle? Angle, theta? Oh, okay. Theta equals A over R. Arc length, 100 pi over radius. I guess, oh, do you folks know what a pendulum is by the way? You know what a pendulum is? So a pendulum like a swing. Oh, let's simplify this, by the way. What is 100, sorry, 10, well, 100 over 150. Two pi over three. Would that be a fair angle for them to also then, as just an extra added, say, find the cosecant as an exact value? Is pi over three one of our special triangle angles? Now, I know this is two pi over three, but I think if we sketch it, we'd find the reference angle. I'm trying to think, how else would they add one extra step to this? Also thinking that might be a nice one to add to one of our mocks for next year. Now that I think about it, note to self, include arc length and exact value in one question. Ooh, I like, I like. I think there's another arc length question, if you go to number 21. They can ask you arc length on the non-calc section, and remember, your provincial and your practice exams will have non-calculator sections and then calculator sections. I think I said this last time, I'll say it again on Monday after school, next week, which is when your first mock is, when you finish the non-calc section, check your answers, check your answers, check your answers. You'll have 45 minutes, and really it should only take you about 35 minutes, so you'll have about 10 minutes to check your answers, but then don't just sit there, go do the trig identity at the end, because you don't need a calculator for that anyways, and then do the transformations questions, because you don't need a calculator for those anyways. Then if you still are ahead of the curve, in other words, if Mr. Comozzi and myself still haven't said, hey, turn in part one, pick up your calculators, then start the multiple choice, we'll usually try and set it up so that out of the first eight or nine multiple choice on the calculator section, you probably do five of them without a calculator. Certainly a lot of the transformations stuff, okay? You would have figured that out probably, but I've just learned never to assume. Oh, let's look at number 21. It says the circle below has, with center O has a radius of 30 meters, find the arc length. Oh, what's wrong with this picture? Degree, yeah, okay, no problem. A equals R theta, radius is 30, be really careful if they're sneaky and stick a diameter in. By the way, I think 15 degrees is gonna be pi by 12, I think 180 divided by 15 is 12, but anyways, just in case we would go 15 degrees times, this time we want the degrees to cancel, so we want the 180 down there and we want the pi up there. It looks like it's gonna be 30 times 15 times pi divided by 180, right? I'm wrong, where am I wrong? Sorry, A equals R theta, isn't that the angle in radians, 15 degrees? Mr. Deweyck, I'm the one that's daydreaming. Hey, how big is this angle right here? 15, because it's a circle, radius, isosceles. All right, how big is that angle right there, boys and girls? Let's try that again with 150 right there. This is why Mr. Kamosi is along for the ride. I do these little brain mistakes once in a while. Is that correct now, 150? This was 30 times 150 times pi divided by, what is it, sorry? Johnny, what? 78 point, and they said to two decimal places, five, four, did you round off properly? They usually won't be fussy for units, but I'm never quite sure. It may just be you have a table that decides they're gonna be insistent, so let's put the units on meters. No arc length, yeah, meters. What are the units for radians, by the way? Nothing, or you can write the word rad, but the old school teachers will say radians or a unitless measure, because they're a ratio. The units on top of the units on the bottom of the fraction are the same, and so they would cancel, fine, arc length. No, it's just every time some old school teacher is gonna say technically it is, but they're not gonna mark it wrong and prove it. Don't ask me, there are better things as far as I'm concerned to get uptight over than that one. Then we went into graphing trig functions. Grapping trig functions. You need to know the big three graphs. So you need to know, okay, that totally didn't work. Let's try that again, hold the shift key down. You need to know what y equals sine theta looks like. First of all, we talked about amplitude, vertical displacement, phase shift, and period. Now on your test, we asked you to graph a trig function, Johnny. We're not going to ask you to graph a trig function on the provincial, but we will probably give you a graph of one and say, hey, what's the equation? Or we'll ask you all sorts of stuff. First of all, what's the amplitude for sine and for cosine? One, what's the period for sine and for cosine? Two pi. What's the domain for sine and for cosine? All reals. What's the range for sine and for cosine? Negative one, less than or equal to one. What about for tangent? Well, in my class, I call tangent the ugly cousin. I told my students and my family, on my mom's side, she's from a big family. We have one of my cousins, he's the black sheep. Marvin in and out of jail, shows up on his Harley sometimes. When I was younger, I thought he was pretty cool. Now I go, you're 45, and every two years, you move back in with your parents for a month. Anyways, the ugly cousin. Most families have one. Tangent, the ugly cousin. Doesn't fit in, doesn't follow the rules. So if I was graphing y equals sine theta, I think the graph looks like this, except I should probably be a little more symmetrical. I can't drive this to save my life. But here's what I do know. What value is that right there? Two pi. And now everything else falls into place. What value is that right there? Pi. What value is that right there? Pi by two. What value is that right there? Three pi by two. And how high, how low? One and negative one. What about cosine? Cosine, take this graph and just move it right to there. In fact, it was a phase shift of sine as it turned out. What about tangent? Man, weirder, little yuckier. You'll notice sine and cosine, I usually just graph to the right, because that's the way I think of it. Tangent, I usually graph kinda halfway left, halfway right. It's ugly cousin. Here's what I know about tangent. I have memorized that it goes through zero, zero. I know where it's undefined. I can't take the tangent of, I'm gonna swear, sorry, 90 degrees. Ah, how many radians? I know just from my grade nine trig when I was a nerd, I would play on my calculator, tan 90, error. Tan 89, big number, try tan 89.9990. You get a bigger and bigger, bigger number. Don't need to try that now, Zanzibar. But trust me, there is going to be, what did you say it was? Pi by two, undefined. You know where else it's undefined? The other direction, negative pi by two. This is how I've memorized it. And then I remember that tangent looks like this. There's my one curve, goes through zero, zero, looks like that. I can figure out whatever else I need, because now it repeats. But if you'll notice very, very carefully, how long is the period, how long is one wave, not two pi? Pi, see it? Half pi that way, half pi that way, that's a whole piece of pi. It's undefined at pi over two, and it looks like every pi past that. So three pi over two, five pi over two, and seven pi over two, oh, and negative pi over two. I can get my domain and range or whatever else I need to. The only other points they'll sometimes ask about is this little guy right here. Or this little guy right here, because that's a special triangle. Did we not have a triangle with a pi over four in it? Which triangle was that? One, one, root two, what was the tangent of the one, one, root two? They might just want you to notice, by the way, that's gonna be one high. And that's gonna be negative one high. That's really how I memorize and remember tangent. Almost everything else I derive using my transformations stuff. So let's see what they can ask us. Oops, I'm going the wrong way. Jeez, I'm a little bit puddle there. Number three on page 20, on page 20. Number three on page 20. Find the minimum value of that. First thing I ask is, which trig function are they talking about? Sign, cosine, or tangent? Sign. Normally, sign goes one down and one up. How is this graph been mucked around? What's my vertical displacement, Johnny? So it's been moved three down, right there. And then what's my amplitude, Johnny? Five, so from three down, from there, from negative three, go five down, five up. So if I visualize this, one, two, from negative three, go five down, five up. You know what the lowest this graph gets is? Negative, by the way, what's the highest this graph gets? Positive two, I think? What's the range? Negative eight less than or equal to y, less than or equal to two. Just trying to think of what variations they could do with this. Anyhow, maximum, oh, they want maximum value? Two. Minimum value, negative eight. What if they asked me to graph this? You know a little alarm bell would go off, Johnny. I got you there. You know why? This is screaming at me as soon as I see this. My students, why is this screaming at me? What don't I like about this? Yeah, it's not factored. They're trying to make me think that this is a phase shift of pi by two to the left and it's not. I would need to rewrite this little middle section here, pull out the two. I'll have an x, I'll have a plus. Yeah, it's a phase shift of pi over four to the left. That'd be a great question, by the way. What's the phase shift, okay? Got you to jump there? Good, now you remember. My kids, it's just, they're just, let me into it now. Did you jump? Yeah. They have the emotional scab there. Nothing will scare them. Which someday when they're a dark alley and a mugger jumps out at them and they just kind of have to think about it for a while and say, oh, I guess that's not to do it scaring me. I'll probably have actually endangered their lives but that's okay. It's still kind of fun. You understand, my students, I've permanently damaged your fight or flight response, right? It's a little Pavlovian conditioning. Number four. Which value for D would result in the graph of y equals four, cosine two x plus D having no x intercepts? C, convince me. We want no x intercepts. Now, Johnny here said four. Johnny, I think that would do this. It would move everything four up but your amplitude would be four. I think that would be a graph that would go, nyu, touch double root right nyu, touch double root. I think you'd have not double roots because these aren't polynomials but I think you'd touch right there. You would have x intercepts there. I think four isn't quite enough. You know how high I'd have to move it? Fine. Yeah, but you didn't know a lot of things. You know everything? Do you? As soon as you said that, I saw you, oh geez, Duke's gonna throw something at me. I better take that one right back. He's gonna come up with something from left field. This is for posterity. Okay. Now, these three graphs, the sine, the cosine, the tangent graph, also allowed us to do exact values for when my graph was right on, when my angle was right on one of the four axes. Screencast the trig lesson. That means I gotta press pause soon. I'm gonna sleep for another 10 minutes and then we'll keep going. No problem. If you need to go, that's fine. Just put the back in there. I figure I got another 45 minutes, but yeah. By the way, I'll let you know what the upcoming topics are and if you're good on them, you can certainly say I'm fine. I'm gonna be doing quadratic equations just after this, word problems, and then I'll do identities last because I think most of my kids found identities okay. Number five. What's the exact value of the cotangent of pi? No idea. I don't even know cotangent. Ooh, what does cotangent go with? Sorry? Tangent. So you know what? I would do a sketch. I would do a sketch of tangent. Oh, tangent goes through zero, zero. At pi over two, there's an asymptote. And at negative pi over two, there's an asymptote. Thank you. Then it repeats itself. So pi over two, there's zero. There's pi. There's three pi over two. There's an asymptote. Okay, it would look like this. Johnny, you said undefined right away. Did you? No, you didn't. Take a look. How high is tangent at pi? Do you see I was able to derive that just by memorizing this one? So if tangent is zero, what will cotangent, the reciprocal be? What happens to reciprocals that are zero high? What do we have right there? Asymptote, so undefined. Ooh, nice. Blah, blah, blah, blah, blah. Number seven. The range of this function is negative k less than or equal to y, less than or equal to 2k, where k is positive. Find the values. Make this a little smaller, apparently. Little spit, there we go. Find the values of a and d. Oh, I have no idea how to do this. I'm not gonna freak out, though. Gotta try something, and I've done enough work to know it. Probably I can figure this out. You know, why don't I write the range of this and then just kind of see if I can match? What's my vertical displacement here, Johnny? D. What's my amplitude? So I think the highest I get is d plus a, and the lowest I get is d minus a. Does that make sense? So if I match that there and that there, I think I have this. d minus a equals negative k, and d plus a equals 2k. Let's just try getting the d by itself here. And maybe if I stick it into there, I might be able to get an expression for d. Good old grade 11 substitution. Plus the a over, right? So what's d the same as? Wherever I see a d, what can I write instead? Okay. Negative k plus a plus a equals 2k. It looks like I can simplify this into negative k plus 2a equals 2k. And if I plus this k to this side, I get 2a equals 3k. It looks like a equals 3k over two. That's wrong, that's wrong. Oh, by the way, I'm already leaning towards this one because I doubt they'd both be 3k over two. I think that'd be a tremendous fluke, but let's prove it. Let's do the same thing, but this time, I'm gonna get the a by itself over here and plug it into there. If I get the a by itself, I'll have a equals 2k minus d and plug that into there. I'll get d minus bracket 2k minus d equals negative k. Ooh, I could be wrong. I see there might be a three popping out of here after all. I would go chunk, chunk, d minus 2k plus d equals negative k. Looks like I get 2d equals plus that over to that side. Ooh, positive k, I stand corrected. I'm not getting a three out of here. I get d equals k over two, which I kind of figured was the right answer. The answer is c. Is that okay? Coming back to you. I'm gonna press stop right now and I'm gonna start the video again.