 So, here's our quiz. You should be able to do all of this no calculator. I'm going to say in your head what I really mean is without the AIDA mechanical device. Hello? Okay? So, let's start. We're going to be flogging those exact values, special triangles, and a few other things. Let's see. First of all, convert the following to radians. Leave your answer as an exact value in terms of pi. So, you can multiply by the conversion factor. The conversion factor says pi radians is the same as 180 degrees. And I think it's fair for me to ask you to actually do this without a calculator, because for starters, yo, the zero is going to cancel and it's really 30 over 18. What goes into 30 and 18? Hey, six does. Divide by six, you get five. Divide by six, you get three. And you end up with five pi by three. Now, that's one way to do it. There's an intuitive way. That's the answer. There's an intuitive way to do it. And it's to say, look, how many sixties is 30? It's five sixties. And I know 60 is pi by three. So, how many sixties is 300? I think it's how many sixties is 30. How many sixties is 300? It's five sixties. So, 300 degrees should be five pi by threes, which works nice for nice angles. Negative 120, see, I could use the same trick. I could say that's two sixties. It's going to be negative two pi by threes. But let's prove it. Times by pi over 180 degrees. Oh, and also the degrees cancels. The degrees cancels. The zero cancels. And I have negative 12 over 18, which I'm pretty sure is negative two pi by three. Going from angles to, from radians to degrees, well, I'm pretty sure it's going to be the same answer as number one. Brilliant, Mr. Dook. Because five pi by three, I was going to say I know pi by three is 60. It's going to be five sixties. Or I can always do it the mathematical way, which is going to be to multiply by 180 degrees over pi. The pi's cancel. And I would say I got a three on the bottom, a 180 on the top, that's a 60, and a one. And I get 300 degrees. A little more interesting, negative four pi by five. Again, I could say, well, I know 180 is pi. What's one-fifth of 180? That's a good question. What is one-fifth of 180? 36? I think so. It's going to be 436. Is it negative 144? Let's prove it. Times 180 degrees over pi radians. The pi cancels. I get a negative. 180 divided by five. Yeah, it is 36. I know my five times table. And I get four times 36. It is negative 144 degrees. I think the first question on your test is going to be either go from radians to degrees or degrees to radians, multiple choice. One more for each of those. A circle has a radius of 1.2 centimeters. Find the length of the arc subtended created by an angle of 75 degrees. Why this looks like a job? For the arc length formula. Sandily, I can't remember the arc length formula. It would be wonderful. It was some dumb, easy way to remember what the arc length equation was. Is there? Oh, it looks like the word arc, really? I mean, if I write that and then oh, yeah. I do have to remember that the equal sign comes after the A, but it's arc length, which is the first word by itself. I don't know. Whatever you want to come up with here. Oh, but there's one more thing I have to remember. Trevor, what can you tell me about theta? Can I put degrees in? No, I cannot. I'm going to have to do a calculator. This is going to be 1.2. Now, this one I'm going to have to use a calculator for. If I give you an arc length equation on your test, it's going to have to be nicer numbers than this. Theta is going to be 75 times pi over 180 degrees. Did I say leave your answer as an exact value in terms of pi in the question? Then I guess we should go to our calculator. 1.2 times 75 divided by 180. Oh, works out to a half. So this actually works out to, if you left it as an exact value, I missed, I know, if you left it as an exact, thank you, I've been obvious. If you left it as an exact value, I think it works out to pi by 2. If I multiply by pi, well, pi is 3.14. I think pi divided by 2 is 1.7. Is that right? No answer. Let's try that again, Mr. Dewick. 0.5 times pi, not times 0.5. 1.57. Did I say 1.7? 1.57. Good gosh. Either of those answers is fine. I didn't specify. 1.6, yeah, because I didn't say how many other places to go to. I will on a test. Absolutely, you can expect a couple of number 4s on your test. There will be multiple choice, almost certainly. I'll give you the angle and I'll say, tell me the exact value. In fact, you know what? One will probably be a positive angle. One will probably be a negative angle. Oh, and one of those will probably be a reciprocal as well as a standard sine, cosine, or tangent. So one will be sine, cosine, or tangent. Cosine of 7 pi by 3. In number 4, in all of number 4, Mitsu, I know the angle. So I start out by actually drawing the angle. 7 pi by 3, there's 6 pi by 3 all the way around. 7 pi by 3 is right here. And so I already answer is positive. I'm not going to write a positive sign. Alex, I only write a negative sign if it's negative. And my reference angle is pi by 3. Do I have a triangle with a pi by 3 in it? Why? Yes, I do. It's the 1, 2, root 3 triangle where this angle right here is pi by 3. And what is the cosine of that angle? Adjacent over hypotenuse. The answer to 1A is that. Now Eric, it's possible to check this on your calculator. Mode, make sure I'm in radians. And if I go cosine of 7 pi by 3, I get one half. Most of these are yucky decimals, though, or even check with your calculator. It's kind of tricky to tell if you've got the right answer. Oh, let's have your calculator on your test. No, next test is non-calc. It'll be the last non-calc test of the year. The remainder of them will all be calculator based. B, cotangent of negative 5 pi by 4. Okay, let's do a sketch. There's negative 4 pi by 4. I think negative 5 pi by 4 is that much further. C, A, S, T. I can tell you that the cotangent is going to be negative. And if this is multiple choice, I'd cross off any positive answers. My reference angle is pi by 4. Do I have a triangle with a pi by 4 in it? Well, yes, I do. The 1, 1, root 2 triangle where both angles are pi by 4. I usually just color in the bottom one. Tangent is opposite over adjacent. The cotangent will be adjacent over opposite. It's 1 over 1. Oh, you know what? The cotangent of negative 5 pi by 4 is negative 1. C, cant of 11 pi by 6. Okay. 12 pi by 6 is all the way around. 2 pi. So I think 11 pi by 6, right there. C, A, S, T. C, cant goes with which function? Oh, it goes with cosine. Cosine is positive. This is going to be positive. My reference angle is pi by 6. Do I have a triangle with a pi by 6 in it? Yes, I do. Exaggerate it, Mr. Dewick, so you don't draw it sloppy. 1, 2, root 3. Pi by 6 is this angle down here. Now I need to figure out what the C, cant is. Well, I know cosine is adjacent over hypotenuse. So C, cant, 11 is going to be hypotenuse over adjacent. I think the answer is 2 over root 3. And we're allowed to leave it like this. In Alberta, and since the book we're using is from Alberta, they would have rationalized the denominator. They would have multiplied the top and the bottom by root 3 over root 3. They would have said this was the same as 2 root 3 over 3, which is also fine, but that won't be what you have to pick from. Cotangent squared. What this is saying is find the cotangent to 5 pi by 3, get an answer, and then square it. Okay. 5 pi by 3. Well, 6 pi by 3 is all the way around. 5 pi by 3 is right about there. C, a, s. Now cotangent is negative there, although since I'm going to square it, I'm pretty sure I'm going to end up with a positive answer. The reference angle is pi by 3. I need to triangle with a pi by 3 in it. Conveniently, I have one drawn right here because this top angle is pi by 3. Cotangent, well, tangent is opposite over adjacent, so cotangent will be adjacent over opposite. It's going to be negative 1 over root 3 all squared because it's cotangent squared. Justin, what's a negative squared? What's 1 squared over? What's root 3 squared? Just plain old. Works out to one third. The bulk of your written section on your test is going to consist of a couple of things. It's going to consist of primarily solving trig equations, where you don't know the theta. Okay. Cosine of theta is 1 over root 2. So this is exactly, it's very similar to this, Emily, but instead I'm giving you this answer and saying what could have gone there. I'm giving you this answer and saying what could have gone there. I'm telling you what it worked out to and I'm saying what was the angle. There's going to be almost always two answers. Cos theta equals 1 over root 2. I have to do a sketch because as it turns out, cosine can be positive here or here, and it does say that cosine was positive. I need to find my reference angles. They're the same. How can I find my reference angles? Well, I look at my answer, which is 1 and root 2, and I say to myself, self, do I have a triangle with a 1 and a root 2 in it? Well, yes, I do. 1, 1, root 2, where that angle there is pi by 4, and that's the angle that has a cosine of 1 over root 2. My reference angle is pi by 4, which means that this angle is pi by 4 and this angle is pi by 4. My first root, my first answer is just going to be pi by 4. My second answer is going to be there. Well, this is 8 pi by 4, 2 pi. I'm pi by 4 shy of that, so I think it's going to be 7 pi by 4. Isabel, how many answers did we find? How many marks was this worth? How about one mark for each answer? On your test, I'll actually give you a half mark for that, a half mark for that, a half mark for that, but whatever, one mark for each answer on the quiz. Dominique, this is a quadratic trig equation, yes? So this is the same as a squared equals 3. If I gave you that, how would you get the a by itself? How would you get rid of a squared? Square root, and when you square root both sides, Dominique, what do we have to remember? Plus or minus. So I'm going to square root both sides. I'm going to get this, cotent theta equals plus or minus root 3. Root 3 over what? Over 1, because I need 2 for the whole fraction. I'm going to say here is my cast rule. Cotangent goes with tangent, although it's positive or negative. You know what? There's going to be an answer there, there, there, and there. There will be 4 answers. Do I have a triangle with the root 3 and a 1 in it, Dominique? Oh yes, I do. This is a job for the 1, 2, root 3 triangle. Cotangent goes with tan, tan is opposite over adjacent. So cotangent will be adjacent over opposite. Which of these two angles, the bottom one or the top one, has an adjacent over opposite of root 3 over 1? Right there. Adjacent over opposite. How big is that angle, my angel? I totally agree with you. This is pi by 6. So that means this is pi by 6, this is pi by 6, this is pi by 6, and this is pi by 6. I can now get my 4 answers. My first answer is just going to be pi by 6. My second answer, well, all the way around will be 6 pi by 6. I'm pi by 6 shy of that. Dominique, what'd you get? I agree with you, 5 pi by 6. What'd you get for the third answer, my child? 7 pi by 6. And how about the very last answer? How many answers are there? 4. How many marks is this worth? How about a half mark for each answer? Sure. That works for me. Give yourself a score, please, out of 13. Make sure your name and your score is on them. Pass them to them. So we looked last day at graphing trig functions if you would be so kind as to turn to the homework from page... Well, we did actually that one there, didn't we? Page 278. I think for your homework I actually assigned number 4 and then like 8, 9, 10, 11 or something like that. Yeah. Any of these you want me to go over now is your chance to ask me. Then turn to lesson 8. Page 283. What we are going to do, Ryan, is all of the Unit 1 stuff. Now, this is nice in some ways. This is a good review of transformations. So we're going to stretch the graphs. We're going to reflect the graphs. We're going to slide the graphs. It's a bit trickier because these aren't the prettiest, especially tangent, the ugly cousin. These aren't the prettiest ones. That's not too bad. So over the next couple of lessons, we're going to consider graphs of the functions whose equations are that and that. And you'll notice I wrote sine... We wrote sine down. We wrote cosine down. We haven't really written tangent. It is the ugly cousin. We've written it a little bit, Emily, but not as much. Okay? Now, quick refresher. What did a number in front of everything here do? Vertical or horizontal? Vertical. In fact, this is going to give us vertical... Hey, Mr. Duke, you need more room. Thank you. Vertical expansion or compression. It's going to change the amplitude. If I put it in trig terms. What about this? Vertical or horizontal? Horizontal? Why? How do you know? Next to the X? Okay. This is going to give us horizontal expansion compression. This is going to give us a horizontal slide. And this is going to give us a vertical slide, except we're going to give those two tricky names. It's going to be called a phase shift. Right out of Star Trek. Today, we're going to look at the first two. The ones that I've labeled. It says this. Use the knowledge gained from the transformations unit to describe how the graph of the given function compares to the graph of y equals sine x, where x is in degrees. Holy smokes, they swore again. Cross out the word degrees and write in all capital letters with authority and emphasis. Radiance, baby. Radiance. He's persistent, isn't he, Brett? He keeps calling up. But you haven't answered the phone, have you? Good for you. This, too, right here. Vertical or horizontal. Okay. Now, here's what we would have written in unit one. We would have said this was a vertical expansion by two or compression by a half. Let's see. It would be backwards if it was next to the y where it belongs. Is it right next to the y where it belongs? No. So it's no longer backwards. It actually is two times as big. It is an expansion by two. But what we want to do today, Trevor, is put it into trig terms. Normally, what's the amplitude of sine or cosine? It was a nice, easy one to remember. It could have been ugly and it wasn't. Normally, what's the amplitude of sine or cosine? One, if you double that, you know what your amplitude is? This sine graph, if I were to sketch it, would be really, really easy. I could sketch it by being very, very clever by actually just changing the graph paper rather than changing the actual graph itself. And I'm waiting for a program to open up because I hit a button on my keyboard and it's taking its sweet time. I'm stalling. I'm stalling. I'm stalling. It ain't going to happen, is it? So I'm going to have to do this manually. And then I guarantee two programs will open up. There's one. I stand corrected. Maybe the button didn't work. Here's sine. Here's how I told you I always graph it. Of course it was going to open up eventually. What did I put here? Remember? No, I didn't put 360. How dare you? Two pi. Ellen, what did I put here then? What did I put here? Pi by two and then I said, okay, the amplitude is one and negative one. And I told you the sine graph looks like this. Starts at zero, zero. And then it goes top, middle, bottom, middle. Now this is the graph. Pretend that goes right through there, sloppy. This is the graph of y equals sine x. What if they want me to do that? Easy. You're going to find one of the tricks we do, Justin, is we often say it's easier to change your graph paper than it is to change the curve. I'll just change my scale. Does that now have an amplitude of two? There's my vertical expansion by factor of two. What about here? B. Horizontal or vertical? Horizontal. In fact, this is going to be horizontal compression by a half because it is backwards because it's next to the x. Here's how we're going to say this in trig terms. The period which used to be two pi is half as big. The period of this one is just plain old pi. And you know what? I could sketch that one being very clever as well. Let me just erase this and then we fix this back to the way it was where that was one and that was negative one. So there was y equals the sine of x. How would I graph that? What's my period now? Pi? See, I would do this. How long is one whole graph now? Pi. What's going to go right here? Think about it. Do the arithmetic. Pi by 2. What's going to go right here? Do the arithmetic. Yeah, half of a half. Pi by 4. And 3 pi by 4. There, done. Because the sine graph looks the same everywhere. It's just really stretching and shrinking. You know what? Sometimes it's easier to change your graph paper. Not always, but sometimes. What's this going to do here? There's going to be a vertical expansion by 3 and there's going to be a vertical reflection. What do you think the amplitude of this graph is? Ah, it's not negative. Amplitude by definition is always positive. It's always a height. It's a scalar for those of you in physics. So the amplitude is 3. What does the negative do? It flips it. Going back to my original sketch here. Has the period changed at all? I'd still put a 2 pi there. A pi there. A pi by 2 and a 3 pi by 2. What's the amplitude of this one? Of y equals negative 3 sine x. So 3 up and 3 down. But what does this negative do, Dominique? It still started at 0, 0. But instead of going up, what? Down. But then it's still middle, top, middle, bottom, middle, top, middle. It's going to look like this. Not too bad actually. The last one. Here we have horizontal compression by 1 third. And we have horizontal reflection. Your period has changed. It's going to be 2 pi over 3. That's how long one wave is. How do I sketch this? Well, I would clue in that I'm going to go to the left. I'm going to give myself some room to the right. But I'm really going to end up going this way because it's a horizontal reflection. What would I put right here normally? 2 pi. What's the period now? 2 pi by 3. Oh, but there's a reflection. Negative 2 pi by 3. What's that? Well, it's half of this. What's half of negative 2 pi by 3? Negative 2 pi by 6. Half is the same as putting an extra 2 in the bottom. So instead of a 3 down there, you got a 6 down there. What would that be? If this is negative 2 pi by 6s, what would this be? Negative 1 pi by 6s. Oh, negative 1 pi by 6, negative 2 pi by 6. Can you tell me what this one here is going to be? Negative 3 pi by 6. I don't care about common denominators. I'm just trying to fill in the numbers right now. And instead of going this way, going this way. So it's going to start out 0, 0. Has there been a vertical reflection at all in this question? I should have written down the equation, which was y equals sine of negative 3x. Has there been a vertical reflection at all in here at all? Nope. Then we're going to start out going up high, but this way. Up there, there, there, there. As long as that's 1 and that's negative 1. It's going to look like a little neater with graph paper, probably Carly, but whatever. So here's what I want you to notice. What's the amplitude? 2. What's the amplitude if this one, trick question? 1. By the way, what number is in front there? It's invisible. What's the amplitude? 3. What's the amplitude? 1. Let's fill in this chart here. What's the amplitude? 1. What's the amplitude? 2. What's the amplitude? 1. What's the amplitude? 3. What's the amplitude? 1. What's the amplitude? 5. What's the amplitude? Oh, come on. Don't die on me, people. What? 1 third. Let's generalize. Justin, what's the amplitude? A. Conveniently, amplitude begins with what letter? Which helps you remember? Yeah, you know what? This number, which was a vertical stretch, so you know what it still is, but for trig purposes, we call it the amplitude. That number there will be the amplitude. Okay. What was the period? Two pi? Yes? What was the period? Two pi over two. What was the period? Two pi over, I'll say one even. What was the period? Two pi over three. Spot of the pattern, let's see. What's the period? Two pi? What's the period? Two pi? What's the period? Two pi over two, which I'll let you reduce to just plain old pi. What's the period? Two pi? What's the period? Two pi over three. What's the negative do? It doesn't change the period, it changes which way we go. It changes the period. What's the period? Two pi over four, which does reduce to pi over two, but I'll let you do that. What's the period? Well, this would be two pi over one half, and then I would say, how do I divide by a fraction, flip it, and multiply? I think this ends up being four pi, which actually, Brett makes sense, because this is also a horizontal expansion by two. If you're being two pi long, it should be four pi long. But can you spot the pattern, Ellen? What's the generic? If there's a number there, what's the period? Spot of the pattern, spot of the pattern, spot of the pattern. Two pi over b. That's your period, always. And you do want to memorize both of those, but we'll be flogging it to death so much most of you just will. Would you affect similar effects for cosine? Yep. What about for tangent? Ugly cousin. Turn the page. Here's a little summary. It says changing the parameter a on sine and cosine results in a vertical stretch with the following, expansion or compression. But in both of these, a equals your amplitude. Oh, and if a is negative, you also get a reflection. Now, changing a with tangent just gives you a plain old vertical stretch or a vertical shrink. But we really don't look at that one that much. Tangent's the ugly cousin. Changing the parameter b right there, right there, and right there. So now we are going to look at tangent a little bit. It does give you an expansion or a compression, but really the period is 2 pi over b for sine and for cosine. Ah! But for tangent, the period is pi over b because tangent's original period was pi. Oh, Breanna, if we are in grease, if we pick up the phone, 360 degrees over b, 180 degrees over b, but let's not pick up the phone. In fact, here's a lovely summary. The amplitude, why don't they put the a in absolute values, a.k., amplitude is never negative. They said you can get it from the graph by going the top minus the bottom divided by 2. Remember I said to you when we looked at the graphs, how high, how low, how low. What's the total distance? Divide by 2. There's your amplitude. Although it's easier to find from the equation, it's what's in front of the trig function. Period, 2 pi over b. Oh, they also put an absolute value because period is also always what? Positive. The negative just tells you which way to count. Tangent almost as easy except it's 180, 180, and no amplitude, 180 or pi, and no amplitude because it's got a range of all reels. So here's your hints. Sketch the graph and then adjust the graph paper. Now eventually that trick's not going to be quite as useful when we do slides as well. Then it's going to be easier to actually just graph the whole thing right the first time, but we're going to build on this idea of, hey, look, find the period, divide it into four equal chunks by dividing it by two and then dividing those by two. Pick a nice scale, figure out the amplitude and we will go. Oh, and then it's top, middle, bottom, middle, top, middle, bottom, middle. All right, example two. It says consider the graph of y equals 4 cosine of 2x. State the amplitude and state the period. Tyler, what's the amplitude? 4, I totally agree. Now, here's what I would do right away then. I would say how high is this graph getting? How low is this graph getting? Negative four. On my graph paper, I would give myself little guide rails. I would right away put a dotted line right there and I would right away put it, I better count this right one, two, three, put a dotted line right there and I would say, Tyler, I know this. This graph is going to be bouncing like that. Okay. I know one more thing. Which trig function did they give me? Sine or cosine? Cosine. Positive cos or negative cos? Has it been flipped? No. You know where positive cos starts? A pi. Sine starts at zero. Remember the S letter? We said cosine looks sort of like a letter. See what I really want you to realize, Brett, is that cosine starts, there's my first point. What's the period of this graph? How long is one wave? Two pi over B. What's sitting where the B is, Justin? In simplest lowest terms, what's the period? How long is one wave, Justin? That means on my graph, if I go pi further, I'm right there. Again. And if I go pi further, I'm right there. Again. Now, I'm keeping in mind that the cosine graph that I taught you last day, look at this, that's what cosine looked like. Here's my question, Spencer. Between the two peaks, when am I at the bottom? Exactly what halfway? In this graph, when will I be at the bottom? Exactly halfway. I will be at the bottom, right there. Three squares in, since it's six squares wide. Pi by two, but I don't really care. I'm just trying to get them. Okay. Oh. And when will I be in the middle the first time on my way down? Exactly halfway between the halfway. Right, Eric? Three square, you know what? 1.5 squares, right? Yeah, is that okay? This is, we're dividing it up into four chunks. Oh, and I think 1.5 squares this way too. Oh, between these two, when will I be in the bottom? One, two, three, six squares, three squares, exactly halfway. 1.5 squares in, I'll be in the middle. And 1.5 squares past this guy, I'll be in the middle. The cost graph is going to look, it's going to look like that. Now, Carly, sometimes they'll give you the equation and say, tell me the amplitude, tell me the period. Sometimes I'll tell you the amplitude and the period and they'll say, what's the equation? Like an example three. Carly, what's the amplitude in this question? 3a? What's the period? Now the most common mistake kids make is they do this. They do the first part, right? They go, oh, Deweyck says the amplitude goes in the front, which is true, and it says sign, so I'll go sign, which is true. And they want to do this, which is wrong. The period doesn't go there. What goes here is that. What did we say that the period worked out to? What? 2 pi over B. In this little chart right here, where it says period equals 2 pi over B, instead I'm going to get the B by itself. Now, don't write this down just yet. This chart is saying, 2x12, how could I get the B by itself? Could I move it to the top? And then move this to the, this is what I want you to write down. If they ever want you to get the number in front, which is B, that's always 2 pi over the period. Write that down in this chart underneath where it says note. Oh, except Caitlin, tangent is the ugly cousin. If you want to find the number that was in front of tangent, it's always pi over the period, not 2 pi over the period. Period is 2 pi over B. B is 2 pi over the period. Period is 2 pi over B. B is 2 pi over the period. Period is 2 pi over B. B is 2 pi over the period for sign and for closer. Tangent, replace the 2 pi with a pi and it's the same thing. So what does that mean here? The number that's going to go here, B is going to be 2 pi over what Carly said the period was, which is pi by 6. Ryan, look up, get the other part later. Don't fall behind trying to get caught up. Ryan, how do I divide by a fraction? I don't. There's a trick. Well, how do I divide by a fraction? You guys know? Flip the bottom fraction and multiply. This is actually going to be 2 pi times 6 over pi. It's multiplied by the reciprocal. Dominique, what happens to your pies? In fact, what is 2 pi times 6 divided by pi work out to? 12. This is your equation. 2 thirds sine 12 x. This will be a multiple choice question and I guarantee when I put a pi by 6 right there, a lot of kids will want to pick that one. The period never actually shows up in the equation. B shows up in the equation. And Cara, what's B? It's 2 pi over the period. Do some math. A cosine function having an amplitude of 3. Okay. Y equals 3 cos. Emily, what's my period here? I'm not putting a 720 here. B is, well, normally it's 2 pi, but we're in degrees. So what am I going to write instead of 2 pi? Darn right. 360 degrees over the period. What is 360 over 720 in lowest terms? I hope you're getting 1 over 2, yes? What is this in lowest terms, kiddo? Don't the degrees cancel and the zeros cancel and it's 30, isn't it 1 half? Yes? That's what's in front of the x. Tangent, ugly cousin. Oh. What's the amplitude that they mention in C, Alex? That's a great question. What's the amplitude that they mention? They don't because tangent has a range of all reels. So it's just going to be y equals 10, but it is not going to be that. Pi over 2 is not going to show up. What's going to be here? I'll write it down. Don't you write it down yet? Is the letter B, but we're going to replace the B. Well, B is for sine and for cosine, it was 2 pi over the period. Tangents, the ugly cousin. What it's going to be? Pi over the period, which is pi over pi over 2. How do I divide by a fraction, Alex? This is going to be pi times 2 over pi. Oh, what happened to the pi? In fact, you know what B is? Just plain old equation. That equation there has a period of pi over 2. Two more, we're done. Example 4 says, consider the graph shown. State the amplitude, state the period. Okay. What's the amplitude? This is a bit tricky here. It says that's pi by 2 right there. And I think it repeats right there. It's one of the few times that I don't count from peak to peak or from trough to trough because they didn't give me a good grid. But I think that's the length of one wave. How far from pi by 2 to 3 pi by 2? How long is one wave? I can't hear you. 3? Don't think so. How far from pi by 2 to 3 pi by 2? How about just that? Right? Yeah, it's 2 pi over 2. Pi, right? Write the equation of this sine function. Okay. Y equals 3 sine. Got to be a little more careful though with this part. It is not going to be pi x. No, no, no, no, no, no. B is what's going to go in front of the x. What's B? B is 2 pi over the period. Oh, Trevor, what's this in lowest terms? What happens to the pi's here? In fact, you get that equation is that graph. That graph is that equation. Turn the page. Quickly glance at this graph right now. Can you tell me, is this going to be a sine function or a cosine function? I'll just be glancing at it. Why? It starts up high also because it says cosine up there on the graph. But even if they hadn't, by glance at that, I'd say it's going to be way easier to write this as a cosine function. You're going to find out that you can actually write this as a sine function as well, but you have to slide the graph left or right. You can also write this as a negative sine and a negative cos, but you have to slide the graph left and right. I always pick the simplest one. Cos if possible. What's my amplitude here? Sorry? What's my period? How long is one wave? Here they did give me full grid paper so I can go from peak to peak. How long is one wave? Ryan, how long is one wave? You going to wake up? How long? Yes, thank you for getting that. The period is 12. Does that mean I'm going to go like this? No, the period doesn't show up in the equation. What shows up here is B. In fact, you know what? What did I tell you B is? I told you to memorize it. B was equal to what? What shows up is going to be 2 pi over the period. There's your shortcut. If you want to, you can actually stick that expression right into there. Now, 2 pi over 12 does reduce what goes into the top and bottom evenly, Spencer. You can do the calculation over here, Sire, or if you want to, you can just go straight into plugging it in and then sometimes it'll reduce in your head like this one. Sometimes it'll be a fraction. How do I buy a fraction? You can do it right in the equation if you want to as well. I don't care which. There's your sine and your cosine stretches. What's your homework? Number 2, 3, and 4. Five is good. Six is good. These go pretty quick. Seven is good. Eight is good. Mr. Duke, you assigned everything so far yet. Lots of time. Shut up. 10, 11. Mr. Duke, stop it. 12, really? Skip 13. Thank you. 14 is good. 16 is good. Now, let's go back. I assigned right now 12, 14, 16, and then 1 through 11. Let's look at 1, skip B, skip D. 3, skip B, skip D. 4, skip B, skip D. Yeah, the other ones you do in practice. Well, you got 15 minutes. You should be able to get it done pretty easily. Actually, it's pretty quick. I hope.