 So sorry what what we're asked 3b. Oh Okay, by the way remember I said they've rationalized the denominator. This is not the answer. We're gonna get we are probably gonna get Something like this Route three plus one all over to route two. I think that's what we'll end up getting So if you did get that you're right and I said to check All you would do is you would write this as a decimal So the square root of six plus the square root of two equals divided by four equals and then go the square root of three Plus the square root of one equals divided by two root two equals and you'll get the same decimal. So if you got that you're right I'll do it anyways What are the angles that I know in radians that I have special triangles for what are the big three? Okay, so I would make a little list over here. I would say I know pi by three Pi by four and pi by six although Hannah I would be clever because it's over 12 I'd do myself a favor and get a common denominator. This is four pi by 12 This is three pi by 12 And this is two pi by 12 How can I use two of these and get an answer of pi by 12? I can either go three minus two or four minus three I think I'll use these ones because they're smaller numbers and maybe I'll do better I think I'm gonna write this as the cosine of Alpha minus beta Where alpha is three pi by two twelve Except Blaine. I'm gonna write pi by four because that's the angle. We're familiar with Minus and Beta is two pi by 12, but I'm gonna write pi by six because that's the angle we're familiar with Can someone read to me? What is the cosine of alpha minus beta except instead of alpha? Can you say pi by four and instead of beta? Can you say pi by six? It's it's it's coast coast minus sine sine or coast coast plus sine sine I can never remember. Okay, so it's gonna be coast pi by four coast pi by six plus sine pi by four Sine pi by six and I know all these. I just got to draw my triangles. So over here. I would draw my 1 1 root 2 very bad drawn triangle, but I don't care. I got the numbers right and I would draw my 1 2 root 3 triangle and now it's more I need to make sure I know the angles and plug Pi by four is this one here. What is the cosine of pi by four? Now I just saw one suit one over root two I just saw one suit. Yeah, there's lots of room unfortunately to make sloppy mistakes on this, too I got because we're doing about ten calculations, but also Martin. Can you see though? I mean the math nerd in me likes this because if aside from solving equations and graphing This is the entire trig unit in one question Let's keep going here. Let's see Pi by six which is that guy right there. Pi by six coast is root three over two plus Sine of pi by four is one over root two opposite over adjacent hypotenuse Sine of pi by six is one half I'm gonna get a lovely common denominator, which is kind of nice I'm gonna get a common denominator of two root two And can you see I'm gonna get one times root three, which is root three. The plus sign drops down one times one Which is what I'm gonna get root three plus one all over two root two Before I forget let's talk about the schedule. Let's talk about our plan of attack so Today I'm gonna be doing one lesson of applications of trig Monday, I'll be finishing off applications of trig work finish trig I'll also have a take-home quiz for you on Monday on identities Be going over the take-home quiz and I have a great big huge take-home review quiz for you here Going over the great big huge take-home review quiz So right now I'm leaning towards your test either being here or being here I know it's the last day before spring break, but I figure you guys agree with me It's better to do it before spring break than have a week off and trying to remember so so I I mean I hate doing a test the last day of Now also during this time Our senior boys are trying to play some basketball at the BC's just down the road and often if they're playing during the day We take a bus down during the day if I scheduled the test here or here and there was a game during the day I would expect that you would talk to me and find a time to write it during your study block The last thing you want to do is not write it before spring break because I'm doing my report cards over spring break Which means I'd be giving you a zero Temporarily on the test and an eye on your report card and also the whole week off you'd forget most of it So I'm trusting you can be mature. I don't quite know whether it's gonna be here and here this weekend I'm gonna start counting up all the days for all my blocks and I'll send an email out probably Well, you guys I'll probably tell on Monday, but I'll also send an email out also that probably means I'll be doing a tutorial in these couple of days here somewhere Is that okay? I Also feel bad I was trying to get the trig at the beginning of the week because I figure some of your other Academic courses are gonna be trying to do tests here, but you know, I I tried to be as nice as I could I just can't fit it in with my other blocks and I do feel bad But it's just not gonna work the way I want it not unless I rush things and have you learn it not as good I'd rather have you have two tests on one day, but have feel more comfortable about the math test. So Not the ideal but as good as we're gonna get so there's three beat what someone else asked another question I can't remember six Yep I'm gonna do coasts of X plus Y I'll let you try sign of X plus Y and I do like this question. I do like this question I do like this question except to me. This is a bit overkill asking you to do both So Hannah, I wouldn't ask you to do both on a test But let me show you what I like about this question because this is also one that really covers the entire The entire trig unit aside from solving equations Here's what I don't like what letter did they use right there? capital X and Tangent is what over what in terms of the graph Y over X I don't like that. They used X's and Y's and these are also X's and Y to me on the test I guarantee I'm probably gonna use like a capital A and the capital B there or a Greek letter and a different Greek Okay, anyhow Here's the first thing I notice if I do a sketch They're telling me that this first angle big X is between 0 and pi by 2 angle X is right there somewhere Sine cosine and tan are all positive and They're telling me that angle Y is between pi by 2 and 3 pi by 2 angle Y is So it's between pi and 3 pi by 2 angle Y is right there Tangent is positive, but sine and cosine are so Hannah I'm gonna do this one here. So X is alpha and Y is beta Can you read to me what cos alpha plus beta is from the sheet except instead of alpha? Can you say X instead of beta? Can you say Y? cos big X cos big Y minus sine big X sine big Y Now the one we did in our notes they had given us cosine Which trig functions have they given us here? You know what here's what I think then what they've really done I think that for angle X Tangent what did you say it was whatever what in terms of the graph I? Think they've told me that Y is 12 and that X is 5 Could I figure out what R is? How are X Y and R related? Yeah, not on your formula sheet. There's a what I what I've told you to memorize is usually because it's not on the sheet I think if you do the Pythagoras, you're gonna find R is 13. Oh And also for angle Y They've told me that Y is 4 and that X is 3 Except I'm wrong Look up for a second Do you see where they told me angle Y was down there can X possibly be positive? Can Y possibly be positive in fact both of these Were negative and that's what made tan positive because you had a negative Divided by a negative and I think if you use Pythagoras again X squared plus Y squared equals R squared I think you're gonna find that R is 5. I'm assuming you're good on Pythagoras. I'm not gonna re-show that Because we're done the hard part now. Are you ready now? It's plug-and-chug Cosine is what over what in terms of X Y and R And we're talking angle big X so we're gonna use this X this R 5 over 13 Cosine for big Y is gonna be negative 3 over 5 Minus Sign for big X is gonna be Y over R 12 over 13 Cosine sorry sign for big Y is gonna be Y over R negative 4 over 5 conveniently I end up with a lovely common denominator Hanna of 65 and in fact on the top I get negative 15 minus negative 48. What's a minus minus the same as? Yeah, this is really gonna be 48 take away 15 35 over 65 Okay That's a nice review almost all things like I'll be honest on your test. I'm pretty sure I didn't give you tan tan I'm pretty sure I gave you like tan Seekant or something in other words. I didn't give you X Y for both of them I'll give you two and I'll give you two but they might not be identical But you'll still use X squared plus Y squared equals R squared to find whatever you're missing I'll also be honest. I probably won't try and make them work out evenly You might end up with an R of like three root five shut up and deal with it Any others Nine I don't think I signed seven did I you know what 11 11 is much more similar to what I'm gonna Give you on the multiple choice Hanna 11 is just like the one you asked me They want cos of X minus Y again They're unfortunately using X and Y again as variables Which I think is dumb because if we're gonna have X's and Y's and also X's and Y's in our work Could you make it more confusing? In fact, I probably if I was doing this question would cross out I would probably do this Just because I'd get confused otherwise with so many X's that are different X's from the other X's that aren't the real X's You're going to need your graphing calculators today Which means if you don't have one it is important that you fess up humble yourself and come get one from me Oh, you guys are pretty good Can you turn please? to lesson nine Sinusoidal functions What are we going to be doing for two days is we're going to be working on applications of trig functions Applications of trig functions. There are lots Trig functions appear all over the place in nature Today in your homework, we're going to give you the equation Next class you're gonna have to derive the equation and I'll actually do one example with you at the end of this lesson Just for practice. It says this read along with me a function whose graph resembles sine or cosine is called a Sinusoidal function the kids get confused playing because it says Sinusoidal they think oh, it's got to be a sine function actually sine or cosine Which both have the word sine in them by the way our sinusoidal functions So sometimes you'll be trying to get a sine graph when it's way easier to do a cosine graph In fact the most common one we're going to end up using is negative cosine because most applications Start out on the ground and negative cosine is the graph that starts out down low The graph of a sinusoidal function is called a sinusoidal graph Many periodic phenomena have sinusoidal graphs example the time of sunrise as a function of the day of the year What's the shortest day of the year? So December 21st would be down here. What's the longest day of the year? Days get longer days get longer days get shorter days get shorter days get longer days get longer days get shorter days get shorter It's a trig function The height of a ferrous wheel as a function of time ferrous wheel goes up ferrous wheel goes down ferrous wheel goes up But anything that rolls Is a trig function? Hannah the depth of the ocean due to changing changing tides as a function of time. It's a trig function We're going to simplify it here. We're often going to use a period of 24 hours. It's not quite I think it's like 23.7, which is the why the tides don't just repeat themselves all the time all the time But tide goes up tide goes down. How many of you have ever seen like in the movies are in a hospital those heart monitors that beep beep Be it's a it's a trig function heart compresses heart relaxes heart compresses heart relaxes In fact, someone can correct me if they get into pre-med, but I think they actually call it your sinus rhythms Like you hear the word sign in there They're all trig functions all over the place Ever bounce on a trampoline trampoline down trampoline up trampoline down trampoline They show up all over the place So in this lesson, we're going to give you the equation and the next lesson will derive the equation now most of the time They're going to be functions of time the variable instead of x will be t But we're going to use our graphing calculator and we'll just clue in we don't have a t-button we use x okay Not not a convenient t-button t there on your graphing calculators actually means parametric equations, which we don't do It's easier to show you by doing an example it says this the minimum depth D meters of water in a harbor t hours after midnight now I read these carefully because here's what this means if I see a number with the hours next to it That's a t variable and when I look at this equation that's sitting where the X is If I see a number with meters next to it, that's a d variable. That's a y variable The equation is this They've written this step in a bit of a weird order the vertical displacement instead of putting it at the end They put it in the front. I would normally write this lastly as five cos point five t plus twelve But we'll leave with what they've written it. I guess they want us to be a bit flexible So take a look at this. What's the amplitude of this? Can you see it? Five what's the vertical displacement? Twelve I know it's written in the front, but that's the same as having a plus twelve there So here's my question if your vertical displacement is twelve and your amplitude is five. What's the lowest we get? Seven what's the highest we get? 17 So part a wants the minimum And the maximum 12 minus 5 7 12 plus 5 17 total distance 10 Determine the period of the function. Well the period is 2 pi over b Which is going to be 2 pi over 0.5 What's 0.5 as a fraction? Dividing by a half is the same as multiplying by yeah, I think the period here ends up being 4 pi Part C says write a suitable window which can be used to display the graph of the function Here's how we're going to show that we're writing a window We're gonna use a square bracket sort of like the edge of a window and what we're filling in Are good values for x min x max scale y min y max scale. Let's think about this This is my x variable right there, but what is it really measuring according to this question? What does the t stand for? time time in what and How long oh did they give me a suggested domain here? Yeah, I think I'm gonna go from zero to 24 hours Now I could have a scale going up by ones But my graph on my screen is only about that wide if I have 24 hash marks Will I really be able to distinguish them very well? Two so you know what I think even better would be I'm gonna go up by fours and what about a y-window well Why is the depth of water? What's the lowest we get? Seven what's the highest? Now I have a personal preference though whenever I'm talking about depth. I like to see the ground So I'm gonna go from zero And I'm gonna go a little bit bigger than 17. I could go 18, but I'm gonna go to a nice round number How about to 20 that should give us a fairly centered graph Scale do I want to have 20 hash marks vertically? No, it would be a good scale to be a little crowded I think would work. I'm gonna go fives because I can count good by fives in my head and it does go evenly to 20 No wrong really wrong answer for scale Now we want to graph this first thing you better do is make sure your calculator is in radians mine is Go y equals and clear any graphs you have there and we are going to carefully type in this equation 12 plus five Coast point five x close bracket Enter to store it double check to make sure I copied it right. Oh Steph I don't have a t. I got to use an x. I don't have a d a clue and that's why Window I Like my x min to be zero my x max to be 24 scale for Y min to be zero Y max to be 20 Scale five And if you hit graph you should get a lovely tide goes down tide goes up tide goes down tide goes up graph Do you I don't know what did I say Now if you did not get that you must tell me right now because we're gonna be using this a whole bunch and You need to be able to do this on your own So don't pass it to your neighbor you need to learn this because you will need to on the test on your own Type in a word problem equation Pick a good window hit graph and then do stuff with it. I will not help Okay Let's do some stuff with this. So we have our lovely title graph D says What is the depth of water to the nearest tenth of a meter at 2 a.m. What are they wanting me to find in this question? Depth what variable is depth in my equation x or y? Why what have they given me in this question to is that an x or is that a y value? Okay, if they give you an x value, we're going to use trace if you hit the trace button and You type in the x value that they gave you of to and hit enter It will tell you the y value that goes with it. You know what the depth of water is at 2 a.m. 14.7 meters couldn't have oh What if instead of 2 a.m. What if they'd said 2 p.m. What would you type in for your x value? Yeah 14 they don't do that very often they do once in a while So make sure you read time very carefully right you'd have to realize we have to go military time right e a Ship which requires a minimum 8.5 meters of water is in a harbor at midnight by what time to the nearest minute Must it leave to prevent ground grounding. What are they asking me to find here? Time is that an x variable or a y value? That means they gave me a y value. What y value did they give me? 8.5, okay, so if they give you a y value, we're gonna use Y to equals 8.5 and then we're gonna find the intersection if they give me a y value I go y equals and under y to I type in that y value 8.5 Which is the depth of the water that we're interested in because if you hit graph now This is a nice visual picture here Madeline, I think this here's what it's saying The ship is fine fine fine. You know what if it's in the harbor at this time. It's gonna hit the bottom Fine fine fine bottoming out and this question says by what time to the nearest minute Must it leave I think it wants this first value here. How can I find this first value here? Where are these two graphs? What? intersect second function Calculate intersection First curve enter second curve enter guess and I get fourteen point six nine two Hours Sorry for point I get four point six nine two Hours four point six nine two hours got a problem. Oh, I'll wait Yes, this is an hours. What do they want the time in? Okay How do I change point six nine? You get this okay, okay Don't forget to enter after you've made your guess or moved it close to the root Back to my question. Hey, how do I change point six nine two hours into minutes? Is it? Convince me. So here's in case you forget I would go like this One hour equals 60 minutes. What's a half an hour? What's a quarter of an hour? And I would say how can I change each of a one into a sixty a point five into a thirty a Point two five into a fifteen and hopefully you clue in yeah multiply by sixty so if you blank out Don't freak out So what I'm gonna do is this I'm gonna go clear clear clear. I'm gonna go point six nine two times sixty I Get forty one point five Four forty one point five although it says by what time if I round this up to 442 we're actually too late, aren't we? It's already bottomed out. So even though technically I should round this up. I'm gonna be safe and I'm gonna say equals 441 a.m. Better be gone by then or you're bottoming out nope In all honesty, this is probably multiple choice f What's the next time to the nearest necessary minute that the ship can return to the harbor then? isn't it Because now the water is deep enough again So find that oh and then change the answer to hours and minutes as opposed to decimals I'll do it up here slowly see if you can do it on your own I got 752 actually I got 752 and I started to round off properly and I went oh wait a minute It's actually a little bigger than 752. So at 752 it's still scraping the bottom. I better round up even though mathematically I shouldn't situationally here. I better Right. Here's the problem at 752. It's still not deep enough It's it's gonna scrape 753 is when it can get in safely. That's why I think it said to the nearest necessary minute I think what they're saying is oh and by the way, don't just blindly follow math rounding rules Look at the real-world situation. There are some times where you always round up Okay, yes for the page in a certain town in BC The time of sunrise for any day Can be found using this equation What does the letter t stand for according to this question? Yeah, time of day, baby, right time and hours after midnight the t stands for the time of the day What does letter d stand for? the number of the day in the year so day 34 would be February 3rd, I think if I've done my math correctly right day 119 would be I have no idea but somewhere in January February just past March somewhere into April right a suitable window What's sitting where the X's? Day, I guess I'd like to start out with day zero Which of my final day beat? 365 by the way take a look can you see b is 2 pi over the period? I think the period is gonna be 365 good scale You know what why don't we go up by 30s because that's roughly by months not quite but close Close enough. What about my y values? What's the vertical amplitude of this graph? Hannah 60.3 that's the middle by the way, that's also spring and fall sunrise time because that's the middle What's the amplitude? 1.79 you know what I'm gonna keep it simple. I'm gonna call it two Two down from there and two up from there. What's the lowest we're gonna get? 4.3 and you know, but you know what I want the ground I'm gonna go from zero even though 4.3 is the lowest this thing is gonna get Up to 6.3 plus two just to keep the math simple is 8.3 You know what how about from zero to nine that should give us a nice centered graph Scale yeah, I probably go up by ones Now we're gonna carefully carefully carefully carefully type this in Making sure we're not missing any decimals along the way y equals clear what you got clear what you got negative 1.79 Sign 2 pi Bracket oh, I don't have a d. Oh x Minus 78 Close off that bracket over 365 Close off the trig function Plus 6.3 Not gonna hit graph yet because my windows are terrible window Zero to 365 going up by 30s zero to nine Going up by ones And now if you graph you should get exactly one full wave because the period I think is 365 You get exactly one full way. Oh very nice B It says use the formula to determine to the nearest minute When the sun rose on may 7th the 127th day of the year so this 127 Is that an x value or a y value? trace 127 enter 4.963 hours Oh, but this doesn't want hours. What does it want? To the nearest what okay, so I'm going to take the decimal portion the 0.963 0.963 times 60 I'm getting 558 approximately 558 a.m That's what I'm getting 458 can't read my writing. Sorry. Let me I'll make that a little clearer for you guys see Determine on which days of the year the sun rose at 7 a.m. Okay 7 Is that an x value or a y value? 7 a.m. Ah time this time is the y. Yeah, you know what I'm going to use y2 is 7 What if it had been 7 p.m. What would I type here? 19 and honestly the highest this graph is gets is 8 We wouldn't have any solution because the sun never rises at 7 p.m Does rise at 7 a.m. How many times? twice let's find those days You get day 55 you get day 284 It's your homework Number one Cross out f Number two. Let's ooh top secret satellite. I like that number three abc four and then Remember the biblical review of epic proportions This great big thing here This thing Going to give you there's a few more questions you can now do you can now do any word problem where they give you the equation like Number five circle that one number 17 48 107 Take home quiz for you monday I said I was going to do an example of one where we actually found an equation from scratch I'm going to hold off because you guys are looking kind of well Down dead out of its zoning so work on the homework instead