 So, questions from the homework number two, first thing I'm going to type in the equation. Now I got to read this carefully. It looks like time t is the time of the, I think in minutes, and d is, looks like it's the distance north of the equator, looks like it's positive, and south of the equator is negative, I think. So, how far north or south of the equator, well first of all, let's get my calculator up and running. Let's pick a good view window, clear what I have here, make sure I'm in radians, I am, and I'm going to go 5,000 cos pi over 35 times bracket x minus 12, close bracket, close bracket. I think that's right, two open brackets, two closed, yep. View window from zero to, now t is minutes. So, what I'm going to do, I'm not sure how many minutes they want. I notice 12 minutes after it's launched, I see 20 minutes here. You know what, just to hedge my bets, I'm going to go to 40 minutes and I figure if I need more than that, I'll just change it, okay. So there the window wasn't quite so obvious, Victoria, I just scanned the question for anything that had the word minutes next to it. Isn't it what absolute value? No, don't use the square brackets, they're just using square brackets as type font. First of all, absolute value would be a vertical line and a vertical line, which it's not, by the way. Yours is a straight line, oh, that's a typo, it's not, I don't think, this is what it should be, otherwise it couldn't be south of the equator, which it will be I think for one of these questions. I mean it'd be kind of nice, but that also wouldn't make sense, satellites, I'll show you a picture later on of the sunlight hitting the earth during the day and you'll see it is actually a perfect sign curve, you'll recognize it right away. Let's follow the same thing because the way the earth is tilted and on its axis and things like that, okay. Anyhow, okay, then does that answer your question? Let's keep going. I'll pick a scale of about five. Why min? Well my amplitude is 5,000, so I'm gonna go from negative 6,000 to positive 6,000 and a scale, I don't know, 500, okay. No vertical displacement, so here's the satellite. Well I haven't done one whole period, in fact pi over 35, if that's to be, I think the period is every 70 minutes, but I'll tackle that later if I need to. So how far north or south of the equator is the launch site, answered to the nearest kilometers? I think what they're saying is at time zero, because that's launch, what's our launch, how far north are we? We're launching 2,369 kilometers north of the equator at time zero, so I'll make a little note here. Time zero, 2,000, what did I say, 2,369 kilometers. Is the satellite north or south of the equator after 20 minutes? Minutes is an x value, so again I'm gonna use the trace feature. It's north of the equator. In fact it doesn't get south of the equator until I'm guessing 35 minutes in. Oh no, I got a phase shift, Mr. Dewick. Probably 35 plus 12, probably 47 minutes in. Let's see, I'm just curious, 47, am I zero high, oh, I didn't do that kind of a domain. Well, whatever, I'll figure it out later. So what did I say, north, oh what was the distance to the nearest kilometer, 20 minutes in, 3,765. When to the nearest tenth of a minute will a satellite first be 2,500 kilometers south of the equator, 2,500 kilometers, now this is a y value. Now for me to go south of the equator, that's gonna be negative 2,500, and it's a y value, so I'm gonna use y2. And I'm not sure my window's big enough to, oh it is, I was a little worried that I might not quite catch it, but I did right there. I'm looking for the intersection. Second function, calculate intersection, first curve, second curve, since it's only one answer I just hit enter for my guess. Looks like 35.333, 35 and a third minutes in, I even know that a third of a minute is exactly 20 seconds, 35 minutes and 20 seconds in if I really really wanted to be fussy. Now I want to the nearest tenth of a minute, 35.3 minutes after launch, is that okay? Any others? So you guys have absolute value brackets? Okay, I'll fix that for my other classes. That must, you know, I think that's just this addition because the kids have never mentioned that before, so it must have been, I wouldn't be surprised if they changed the font, it could even be something as simple as now they're on Windows 2000, Office 2007 or 2010 when they're publishing. I'd mention it to them. Well, yeah, Math 12 is still going next year. I'll email them. I'll get it fixed for next year. Any others? The tricky part if you have the equation is interpreting what they want you to do. Look at the units. They will always tell you what the variables mean and kind of figure things out. The trickier part is actually getting the equation. Turn the page. Next lesson, please. Lesson 10, modeling trig equations, it says. This is where they're going to tell us about the graph and they're going to ask us to find the equation. And when I do this, there are several key things I'm looking for. And usually this is the order that I look for them. We're going to jot these down. What we're looking for is the following. We're looking for, whoa, I'll talk to you later, the highest point or the maximum. So what words, top or maximum or highest or words that say, look at the highest, I do that. And I always do a small sketch. I look for the lowest point, the minimum, the bottom. If we're lucky it's the ground, in which case it's zero, but not always. Once I know the highest and the lowest, I find the middle because the middle is my vertical displacement. If I know the top and I know the bottom, I can find the center. Then I try and find one point. Now that one point will either be at the top or the bottom or the middle. If it's at the top, which graph starts out at the top? I'm going to be fussy, which cos, positive cos, which graph starts out on the bottom? Positive cos, which graph starts out in the middle, sine. That's what determines which trig function I'm going to use. And I'll phase shift over to whatever this point is if they give me an x-coordinate. Usually it's time, if they say, for example, after three seconds you're at the top, I would say, oh, I'll do t minus three, positive cos. Usually we'll use negative cos because on most of the applications at time zero, we're starting out on the bottom, in which case negative cos, no phase shift is the best. And then the last thing I read very carefully, I find the period, and b is two pi over the period. And you'll just have to read the question. They'll say it goes around every 35 seconds. That's your period. Well, let's see. Example here, warm-up number two. A sinusoidal wave shown has a maximum value of 50, top, a minimum value of 10. So the first thing I would say is this, if you know the top is 50 and the bottom is 10, what's the total distance between 10 and 50? 40, what's the amplitude? What's the middle? What's the vertical displacement? 30. Now, it says, write this where A is positive. I'm going to cross this out because there's no way I would write this where A is positive. I would say, you know what, my graph starts in the middle and goes down. If I want to keep this simple, that's going to be a negative sine graph with no phase shift. What I really need is the period. Well, here's the bottom. Here's the bottom. How long along the x-axis is that? You have to do a little bit of counting, sorry. One, two, three, four, five, six, seven. You get eight squares. What's one square worth? Sorry, four. So eight squares times four. How long is one wave? 32. And I get really lazy. I just go like this. That's B because B is 2 pi over the period. Now, if this is written, I'm good with that. I'll type it into my calculator. If this was multiple choice, what would I do in the answer? I wouldn't write 2 pi over 32. What would I write? Pi over 16. Now, they did say they wanted us to write it as a positive sign. A positive sign would start right there. It looks like 16 in. I would go positive t minus 16 in brackets. What if they wanted me to write it instead of a positive sign? What if they wanted me to write it as a positive cos? I would start right there, which looks like it's 24. t minus 24. So this is, oh, and they want me to write h of t. I'll use their variables. If you're in a rush and you don't use h and t, if you wrote y and x, we would give you full marks. But I would just be thinking bad thoughts about you and be thinking, read the question. So here's the first example. A nail is caught in the tread of a rotating tire at point N in the following sketch. And as the tire rolls, the nail goes up. The nail goes down. The nail goes up. The nail goes down. It's a beautiful sinusoidal. Remember, sinusoidal means sine or cos function. The tire has a diameter of 50 centimeters. And it rotates at 10 revolutions per minute. After 4.5 seconds, the nail touches the ground. It says, use the information to write a scale for each axis. We're going to go straight to the graph. And what we're going to do is we're going to do a sketch. We're going to write the equation. We're going to do a sketch. Now, because they gave me a sketch, I'm going to use theirs. But if I was free-handing, I would do this. What's the diameter? I want highest. How high? 50. I want lowest. How low? Kind of a dumb trick question. 0. I want middle. OK, because I can fill in a bunch of stuff already. H of t equals, what's the amplitude of this graph? 25. What's the vertical displacement of this graph? Also happens to be 25. Now, they've told me they want me to write this as a positive sine graph. It says write it as sine, where a is positive. So I'll write sine. Do I need to phase shift? Where does sine begin? In the middle and goes up if it's positive. You know what? I don't need to phase shift. It's kind of nice. Then it says it rotates at 10 revolutions per minute. After 4.5 seconds, the nail touches the ground. Both of these are giving me the period. Right here is 4.5. What fraction of a wave is that? Three quarters of a wave. So I could figure it out from there. But I'm going to use this. If it takes 60 seconds to go around 10 times, if it takes 60 seconds to rotate 10 times, how long to go around once? Think about it. 60 seconds, 10 times in 60 seconds, how long to go around once? Now, my physics 12s are going, yes, and this is the frequency 10 over 60. So the period is 60 over 10. Take the reciprocal. But I can't do that in math 12, unfortunately, because they don't necessarily know that. So I'm going to say, you know what? Yeah, the period is 6 seconds. I'm going to put 2 pi over 6 and no phase shift. C asks, how far to the nearest 10th of a centimeter is the nail above the ground after 6.5 seconds? Now that we've found the equation, we're going to go to our graphing calculators. Going to clear whatever I have. And I'm going to carefully type in this equation, 25. Sine 2 pi x over 6, close bracket, plus 25. We have to use letter x instead of letter t. No problem. Is that OK? View window. My x is time. It starts at 0. How far over do I want to go? Well, that's 4.5. You know what? I'm going to go to about, well, what's the period? How about 12? Two whole waves. There's no real right answer. Wrong answer would be like 2 seconds. That'd be silly. Let's go to 12. Let's get two whole waves. And I think I can go up by 1s doing that. What's my minimum y value? 0. What's my maximum y value? 50. I usually go to 60. I usually go a little bit higher. Scale 5s would make good sense, I think. Graph. Nail goes up. Nail goes down. Nail goes up. Nail goes down. By the way, to see if I've done this equation right, remember they told me after 4.5 seconds we're on the ground? I'm going to go trace 4.5 seconds. Am I on the ground after 4.5 seconds? I'm pretty sure I'm right. I don't think I fluked into that. Now let's answer the question in part C. This 6.5 that they're giving me, is that an x value or a y value? x, use the trace feature. Trace 6.5. How high are we off the ground? 37.5 centimeters. We're going to add a part D. When is the nail 1 centimeter off the ground? When is the nail 1 centimeter off the ground? 1 centimeter, is that an x value or a y value? That's a y value that we solved by going y2, 1. Hit graph, and we're going to find where they cross. Kind of tough to tell. I don't know what's there, though. Second function, calculate, intersection. First curve, second curve. Guess I'm going to move my cursor until I know. I think I'm safely to the left of it there. Enter, 4.2 seconds in. Sorry? One second off the ground. Well, here I should have said when is it off the ground for the first time. And that's what they'll say in the first one. I just ran out of room to write. I thought why don't I add a y value question? And I just thought this right now. So I'm next class. I'm going to teach this block D. It's not going to be 1 centimeter, because I didn't think it was that low. I didn't think it through. It's going to be like 20 centimeters, so it's a little higher. And I'll say first. Sorry. Yep, good question. Turn the page. Ferris wheel. For my ESL students, if you look up, this is a ferris wheel. In fact, this may be the ferris wheel. Let me see. That could be the Chicago one. The first ferris wheel. I got to go off on a little rant here. Let's read. The first ferris wheel ever built was created by a bridge builder by the name of George W. Ferris. We named him after him. In 1893, it was for the Chicago World's Fair, and it was spectacular. Huge. He tried to raise money, and he was turned down over and over and over for the building, because they thought there's no way you'll make money. As it turned out, it ran 24-7 the entire World's Fair complete sellout, because nobody had ever seen the like. It says this. The diameter of the wheel was approximately 76 meters, far bigger than anyone that you've ever seen. And the maximum height was approximately 80 meters. Now, 80 meters, can someone go 80 divided by three? Roughly how many stories is that? Because it's roughly three meters to one story. How many? So about 27 stories. Just shy of 30 stories, which in 1893 in Chicago was spectacular. It was the highest thing around. It had 36 carts, and each cart held approximately 60 people. Can someone go 36 times 60? How many went on this ride at once? Yeah, it had a ride capacity of what? So about 2,200 people. Huge ride capacity. My understanding is when he built it, the axle was the largest piece of steel ever constructed in history. And apparently, the entire construction took 1 third of the US eastern steel industry for the previous year. 1 third of the US eastern steel industry was working on this one ride. It was spectacular. Afterwards, sadly, it was dismantled and sold for scrap. But amazing ride. The first one ever built. For the longest time, the largest one ever built. Yeah, the one in. I think I'll allow you to Google and find out. I believe the one in London is taller, but I don't think it has the ride capacity that this one had. It says, if the wheel rotates every three minutes, draw a graph which represents the height of a cart as a function of time. It says, show three complete cycles. To you, I'm going to say, I'm just going to draw one. And do you remember what I said we were interested in? We're interested in highest, lowest, and middle. That's our first goal. So carefully read the question, what's our highest point on the y-axis? It's not on, you have to read the question in front of you. Read the information that they gave you. Sorry, 80. I'm going to go like this, Matt. 80. What's the lowest point of the Ferris wheel? It's not zero. You have to read the question. How did you get four? Oh, they told you the diameter. See it? So if the diameter is 76, the lowest is four. What's the middle, plus four, and then divided by two? I think that's 42 of the middle. Because that already gives me a whole bunch of the equation. Certainly it tells me that my vertical displacement is 42. And it also tells me my amplitude, how far from here to here or how far from here to here. 36, 38, 38. By the way, what's half of the diameter? More specific, what's half of this diameter? 38. If you're dealing with a circle, the radius will always be your amplitude. It was on the previous question, too. 38. So we've done highest, lowest, and middle. Now I want to find one point. It says, assume that the cart is what? At time t, we're right there. There's my one point, because that tells me which trig function I'm going to use. Which trig function am I going to use? What starts up high? I'm going to go positive. Close. I have no idea what you said. Well, they said close, but they didn't say it had to be positive. In fact, I kind of don't like this. I would assume that you're at the lowest point at time 0, because that's where you get on. And that's what you're going to find for most of the Ferris wheel questions. They're going to say you're getting on at the bottom, using negative cosine. It's way easier. Which means, what's my phase shift going to be for a positive close? 0, because we're right on the y-axis. So I can over here go t. And h of t. All I need to get is the period. How long to go around once? Three minutes. Period is 2 pi over b. Sorry, b is 2 pi over the period. 2 pi over 3. I was going to go to seconds, but I did notice that in the graph, they want it in minutes. So here's what that means. Here's three minutes. We must be back there, right? What time is that right there? 1.5 minutes. That must be where I'm at the bottom. What time is that right there? Yes, you're going to have some decimals, but you're dividing by 2. It's not that yucky. Minutes, please. Can someone go 1.5 divided by 2, please? 0.75. Good God. That's when you're in the middle. What about that value there? 2.25. That's also when you're in the middle. Do you see how I divided it into four like we did last unit? You go down, you go up. Because just for practice, here's what I'm going to ask you to do now. h of t equals, we're going to write this as a negative cos, negative cos. Let's see if we can do that. What would the amplitude be? Still 38. What will the period be? Still 2 pi over 3. What would my phase shift be for a negative cos? What would my phase shift be for a negative cos? What will I write here? I'll give you a hint. t minus 1.5. That'll phase shift me to there. Will the vertical displacement change? No, no. There is this graph as a negative cos. Same equation. Sorry, same graph, different equation. Let's write it as a positive sign or a negative sign. We'll just do one of those two, because I'm running out of room. Victoria, would you like to write it as positive or negative sign? Did you say negative sign? OK, let's do that. h of t equals negative, and I'll leave a space and put the sign. Will the amplitude change, Victoria? No. Will the vertical displacement over here change? No. Will the period change? No. The only thing that's going to change is my phase shift. So I need to find where negative sign would start. Well, negative sign starts where? In the middle and goes. Although, Ari, there's always going to be an easier one. I can do whatever trig function they want me to. When in doubt, I'll try and do a phase shift of nothing. We're going to use the blue equation, the cosine one, because it's less typing. Let's go to our calculators. Clear, clear. h of t is going to be 38 cos 2 pi x by 3 closed bracket. Now, you'll notice I didn't put brackets when I hand wrote, but I do have to on my calculator, no matter what. Plus 42. I probably should have put brackets, but if you leave gaps, it's kind of understood where sloppy when we hand write. Enter. Window. Start at 0. How long to go around once? Three seconds. But look at my next question. How many minutes are they talking about here? Five. So if I went from 0 to 3, that'd be silly, because 5 doesn't appear. I'll go from 0 to 6. Two full revolutions. Make sense? Ah. If I was really clever, what would I pick as a scale? No, I wouldn't pick one. Absolutely not one. No, I wouldn't pick 1.5. I think I'd pick, wouldn't I pick 0.75? Look at my graph. Isn't each important point every 0.75? That's what I'm going to do is my scale, because that way, if nothing else, I can glance and line up those points with the hash marks. That'll, I can eyeball it, and I'll know whether I got the right equation. Y min is 0. Y max, what's the highest we get? 80, I'll go 80, I'll go 90. And I'll go up by tens, or fives, whatever. I'll hit graph, let's see. Ferris wheel goes down, ferris wheel goes up. Ferris wheel goes down, ferris wheel goes up. That looks great. C asks, how high is the cart five minutes? OK, five minutes. Is that an x value, or is that a y value? x value, we're going to use trace, trace, x equals five. y equals, holy smokes, exactly 23. That's a fluke. Although if I was really fussy, instead of a y, I'll use the letter h equals 23. I wouldn't take marks off, but let's do it right. How many seconds after the wheel, OK, how many seconds they're asking me to find an x value? After the wheel starts rotating, does the cart first reach 10 meters and that's how I should have phrased the previous question, by the way, as well. I should have included the word first. 10 meters, is that an x value, or is that a y value? You guys missed the fact that I didn't say, because I asked you to? OK, I want this one. Second function, calculate, intersection, first curve, second curve, guess. 1.23 minutes, 1.228, I'm going to write down, because that's in minutes. And they want an answer to the nearest second. How do I change minutes into seconds? Times by 60, so I'm going to go like this. 1.228 times 60, 74 seconds in, 1 minute, 14 seconds, if you really want to be fussy. So on your test, on the written section, I'm going to give you some kind of a word problem. And you're going to have to find the equation. And then I guarantee you, for a part b, I'm going to give you an x, find a y using trace button. And for part c, I'm going to give you a y, find the x using y2, send it across. Put your pencils down and look up for a second. I've fibbed to you a little bit. Now we've used our graphing calculators, and we've graphed for c, we use this trace function. The other thing I could have done is I could have gone in my non-grafting menu, I could have just put a 23, sorry, 5 right there. I could have gone 38 cos 2 pi times 5 divided by 3 plus 42. My argument there, and I think I will get 23, my argument is if I'm going to type that, why not type it as y1 so that I can see if I've typed it right by looking at the graph? It's not extra typing. OK, you do have to set up the view windows, but I think that extra five second investment is really worth it. Oh, and I could have solved this question here as well. I could have solved it by putting a 23 right here, minusing 42 from both sides, divide by 38 to get the cos by itself, replacing with an a temporarily because it is a period change. And I could have solved it by hand like we did for period change. Again, yeah, my comment is if I'm going to do that, why not just type it into there? Because I'm going to be typing all on stuff anyways along the way. It's much faster to use the calculators. So can you do these by hand? Sure. When I said you absolutely have to have your graphing calculator, I guess I technically fibbed. But the bang for your buck there is totally worth it in my mind. What's your homework? Take home quiz. Number one, number two, number three, number four. Now you're going to have to figure out when March 11th is what day of the year it is. But I think it's fairly easy to figure out because how many days are there in January? 31 plus, how many days are there in February? It says this year is a leap year. So there's going to, this year, where am I? Is a leap year. So if there's a leap year, how many days are there in February? 29. And then March 11th, I think that's the day of the year. On a test, I would tell you what day of the year it was, but for what it's worth, doable. Oh, they give you the, ugh. Oh, it's in degrees, forget it. We've left degrees behind. They're gone. Even if they bring chocolates, I'm not taking them back. What's your other homework? Well, let's go to that great big trig review, please. Were you looking for someone? Nope, okay. I'm almost done. Ladies and gentlemen, that's it. That's trig. We have gotten tricky with it. I'm probably going to do a little spiel next class, some more hints for identities because my sense from my three classes as kids are struggling, and I deliberately moved on because I thought you got tired of swallowing water. We went to a different area and we're going to come back to the pool. Don't worry about it. These are the questions that I am assigning for the review that I think are fair game for the test. So I think I've already said two, three, four, five, six, 10, 13, 14. I already said those, yes. I'm going to add 16, 17. Hey, I can now do Ferris wheels. 19, 22, I already assigned, right? 25, 26, 27, that's a nasty because that's a addition identity added to another addition identity. But can you see it is sine alpha plus beta plus sine of alpha minus beta? Ooh, I see a Ferris wheel, 28, absolutely, 33. Now, if I look at 33, here's what I see. I see coast, coast minus sine, sine. I think that's on my sheet somewhere. It's a, oh, you got that from memory? I'm impressed. It is as it turns out. No, he's smarter than he looks. 38, 40, 43, 44, 45. How will I solve 45? Okay, 46, 46 is one of those I told you about. It's an alpha plus beta, but they've used one of those four corners, which means a bunch of stuff is going to cancel and you'll get a lovely, well, I think I told you the answers are almost always positive sine, negative sine, positive coast and negative coast. Although I noticed they put a two X in there just for giggles. 47, general solution practice. 53, 55, 56, ooh, 57. 58 is an example of a nasty curveball. 61 is another solved by graphing. I think I've given you about eight of those. So I'm probably gonna now only pick the ones that look really weird for you to practice. But trig identities, we still need to practice. So 64, 66, 67, 68, 69, 68 was nasty. 73, again, look at 73, it's an addition identity. It's three pi by two, one of our corner angles, it is. Look at your answers. Are they negative sine, positive sine, negative coast and positive coast? A bunch of stuff will cancel, it'll be one of those. 74 is a decimal, I've already done that. Ooh, logs and trig, I'll use that. 76 is a quadratic trig, we need to practice those. 81, skip, skip, skip. And it's a decimal, I've done a bunch of those. 86 Ferris wheel, 87, 88, both identities. 91, 95, 98, we have reached 100. 106, 108, 115, 116, a lovely quadratic trig. Ooh, here we go, a restrictions question, 117. Skip, skip, skip, skip, skip, skip. 123, 124, arc length, amplitude, arc length, exact value, phase shift, 134. Give you an example of another mini-curve ball with an addition identity. Should be on the last page, I think, yeah. Woo-hoo, written. I like number one, I like number one, and I like number one, number one is a nice question. I like number one. And those of you that have me in physics, hopefully you've clued in by now, I'm nuts about amusement park rides. There's a very good chance that your application word problem might be a Ferris wheel. So I'll also give you 10 to try as well, 20. No, it's not all the written, but it will be. Here's the problem. What we did last unit, the written questions that I gave you last unit were, it was solving trig equations, but they were linear, they weren't squares. The written on the provincial is quadratic trig equations. In other words, this unit contains identities, which is always on the written section, quadratic trig, which is almost always on the written section, and word problems, which is occasionally on the written section. 21, now again, are you doing all of these? No. Are you looking at each of these, and if you're saying, yeah, I know how to do that, then you're fine, move on to the next one. I tried to give you way too much, I tried to give you extra practice so that if you wanna prep for the test, you can really work at this, and the test is two weeks away for you guys, basically. So you got lots of time to work on this. My goal is not that you see every curve ball, but you've seen enough of them that you have your strategies so that those one or two weird multiple choice, which will show up, you'll be like, yeah, you know what? I've seen, oh, this is exactly backwards from the one that I saw, or whatever, okay? The class is yours. That's trig.