so let's talk about some trig functions we're going to start with some basic angles I'll show you how to use reference angles to find out trig functions of any angle you ever you come across uh we'll talk about some graphing of some triak functions which is going to be kind of interesting you'll see why they are the way they are and that should be our day so the first thing we're going to start with just a little bit of talk about angles now when we say an angle typically we mean an angle in reference to the the x-axis and we got a couple names for these things our angle would be given here I like to call Ita typically if we're going to start here this would be called our initial side what do they call the the side where the angle ends yeah that's the terminal side of course you see the movie The Terminal has nothing to do with angles but it's a pretty good movie you should check it out anyway now now if we're going to start the x-axis and go counterclockwise which is the the positive angle measurement clockwise or counterclockwise so going this way counterclockwise gives us positive angles if we go this way our counter our clockwise that gives us our negative angles so counterclockwise rotation gives us our positive angles and and in in Converse we have clockwise giving us our negative angle measurements now when we measure angles there's typically two two construct are measuring angles we either do degrees or we do let's talk a little bit about radians versus degrees where do they come from why why we have them radians are kind of interesting they involve that number Pi which really is a weird number in and of itself right I mean it doesn't end it doesn't repeat it's it's the an irrational well actually um yeah irrational number it's it's kind of strange and it's also relationship between the diameter and the circum of a circle and that's that's why we want to incorporate that in dealing with angles because we have parts of circles in that then we have this degree system that we also have uh that that most of the time that's how we describe angles to each other right at least here we say if I say okay can you draw me an angle of 60° you have no problem with that typically say draw me an angle of 3 pi over 14 you go what that's crazy you seriously going to draw well you can I mean you can draw that but you don't have that in our in our heads so normally we like to have ways to convert between them just so we're kind of familiar with them can use both of our angle measurements um if if you're not familiar with this how our radians and our degrees are are connected is with this equation 2 pi radians equals 360° reason why we say they're equal those are both the way you can represent a circle so this represents all the measurements degree wise of a circle and so does this for radians of a circle therefore they have to be exactly the same what this gives us if we do just a little bit of mathematics if we solve for radians or we solve for degrees you can even write like this de it's going to give us a method on how to convert between radians to degrees so for instance if we' uh want to solve for for degrees well we can divide by 360 and that's going to tell us that to change from radians to degrees we're going to be multiplying by pi over 360 that that that worked just fine so here's our our note to convert if you want to go from degrees to radians go from degrees to radians multiply by pi over 180 ulti pi over 180 what that's going to do is get rid of your degrees and introduce to you that that Pi that's going to change into radians for you uh the other way if we want to go from radians to degrees well multiply by what do you think yeah really you're just multiplying by one in a in a special way and if you divide both sides by two you're going to get Pi radians equals 180° so you're just using that equality uh to change between our degrees and radians would you like to try a couple examples to get your heads wrapped around this all right let's change uh we'll just do two examples really quick let's change 200° into radians and we'll change -3i 4 radians into degrees so 200° if I want to change 200° into radians well I'm already in the degree measurement I want to get rid of the degrees and introduce the radians so here I'm going to take my degrees sure in order to get rid of those units degrees you can't have degrees on the top that's going to be degrees squared right that that'd be that'd be ridiculous what we want to do is have our degrees on the denominator of that fraction and that kind of tells you where you which one of these if you if you forget all this which one of these equalities to use which one of these conversions if you're trying to get rid of degrees have degrees on the denominator then your Pi is going to be on the numerator and you'll see that the degree units those are gone you can cancel them out just like you would other variable or or amount can you reduce that that sure go ahead and do that you got calculators if you want to punch in the fraction just press enter what is this going to give us how much 10 10 pi over 9 okay 10 pi 9 radians so right there we know that 200° and 10 pi 9 radians is exactly the same measure of certain angle what other angle I tell tell you we were going to convert oh good number I made top of my head so good to remember that so --3i 4 all it's in radians we want to convert this to degrees which means we want to introduce the degree measurement what what that typically says is well we want to get rid of a pi somewhere so I'm thinking the pi is going to have to be on the denominator of my fraction and if I want degrees at the end of my problem I'm going to have 180° on the numerator my fraction does anything simplify out of this sure four goes in 180 how many times 45 okay and what anything else Pi ah the pies are gone that's great can you do the math tell me what we're going to get how many degrees 135 135 negative negative sure would this angle be measured uh counterclockwise or clockwise could you find the same angle by measuring some distance counterclockwise angles are strange like that right you can go negative or positive and somehow end with the same exact at least reference point not same angle same reference point on that Circle kind of cool by the way speaking of graphing let's go let's go ahead and go over just a little bit how to graph these appropriately so if I were to give you let's try 4 pi over 3 can you graph that on a circle or on an XY AIS yes no you can can you I can can you how do you do it how do you do it you know one circle is 2 you do so I know that this is this is actually pi and this is 0 or 2 pi well this has got to be Pi / 2 and this is got to be 3 pi/ 2 here's the way I always like to graph these things what I like to do is break up each section of Pi into whatever denominator I have that's the way I think is easiest for me so I think okay if this is 1 Pi I'm counting Pi being divided by 3 you see you can split this up and say okay this is four 4 * < 3 so what I want to do is find out where the 3 sections are count four of them and I'm going to have my angle is that Mak sense to you are we awake today I know it's Friday right you're like oh just just 47 more minutes please 37 more minutes please but stick with me here folks this is good stuff this trigonometry right I mean some of you trust me you need a refresher on this believe me so if I want to break this up my pi over 3s I think okay well here's I split like that here's one two three sections of equal value inside of my Pi does that make sense let's do the same thing down here so basically we've broken up each Pi into three parts or each 2 pi each circle into six parts how many are we going to count four we want four pies over three so 1 2 3 our angle is going to end right there you feel okay with that so far you sure now if I ask you to graph -2 pi over 3 -2 3 we'd start same initial side but we're not going counterclockwise anymore we're going clockwise where's it going to end you still break it up into three parts right every every Pi would be broken up into three parts if we're counting -2 that means we're going to start here one two let's call this uh 4 Pi 3 this would be our 2 pi - 2 pi 3 whenever you have that situation where one angle and a different angle go in the opposite direction and on the same exact uh terminal side they're called co-terminal it just means there's multiple ways to measure the same terminal side I could have gone around like 50 times right and then ended right there it's going to be the same exact ultimate ending spot how about we graph let's graph two more let's graph 5 Pi then we'll graph uh these are a little bit easier -7 pi/ 2 so 5 Pi of we're going to start with our our typical initial side that's our zero angle or 2 pi angle marker we've got Pi here Pi / 2 and 3 Pi 2 why don't you try the 5 Pi start at your initial side are you going to go clockwise or counterclockwise why counterclockwise okay so go ahead and do that count however many pies you have right now now we don't have to break this up at all because well there's no denominator so we're actually counting five pies let's go ahead and do that how many times a round am I going to go okay so I'm going to start here and go well here's here's one Pi 2 pi that's a Full Circle isn't it 3 Pi four you getting dizzy yet and then one more that's gonna give us a five Pi look looks like a spring all wound up do the7 pi over which go on your own of course you know this is zero and 2 pi it's already broken up into pi over 2 is 4 that's kind of Ni that's not a oh let's see where we end up okay we'll start our initial side negative I know that means we're going we're going clock I'm going to count 7 pi/ 2s going clockwise that's what our negative says so here's 1 2 3 four five six seven I know we're that as our terminal side we've just gone around time in a quarter time in 3/4s how people were able to end with with exact that cool are there any questions so far before we continue on to some actual trig functions we're going to put this stuff together now no yes you all right so far all right good now of course Somebody went and said how can we relate the sides of a right triangle and a unit circle along with their angles and so we have our unit circle up here and if we were to make a right triangle out of it it's a horrible right triangle there we go if we consider the point XY with any we're being General here so we have any any coordinates how far along is this right here whatever the x is true okay and and how far is this right here and if I say the unit circle how far is the hypotenuse in this case one one good that's that's what's a unit circle unit circle says has a radius of one what we what someone did a long time ago is they say okay we want to somehow represent the sides of a right triangle and call them something we want to call the ratios of our sides of a right triangle some function some trigonometric function so if we have any sort of angle on a right triangle just like we have right here and of course you know that's a hypotenuse we call this is this the opposite of the adjacent okay and this is the adjacent side we had a few trig functions let real clear before we go any further the stuff that says s cosine tangent th those are functions okay you you can't you can't compute those those aren't those aren't a number uh you can't ever just have a say I'm gonna take sign sign by itself doesn't mean anything s has to have some sort of expression of an angle inside of it if we just say can you take sin s plus cosine squar no you can't that doesn't mean anything that that this is not equal to one because you don't have any angle here you have to have exactly the same angle for that to be equal to one you have to have those angles in there for identities to work you to do the computations with them so if you ever find yourself writing s and cosine without any inside part you're probably making a little mistake be real careful that s and cosine have and tangent and all these Tri function have to be associated with some sort of an angle am I getting that across to you these are functions just based on that angle without that that doesn't make any sense okay now let's go back to this let's review what is sign exactly it's a relationship between which two sides good have you ever heard of Chief SOA TOA no maybe in high school Chief SOA TOA he asks you if you get this wrong I'm going to ask you a question yeah soaa says sign is opposite over hypotenuse and cosine is what SOA okay and tangent what's tangent opposite for sure we also get cosecant you know I'm going W down here so you see a little better the cosecant the secant and the cotangent they're just the reciprocals of the original Three tree functions that we have so which one goes with uh the cosine which one is the the reciprocal of our cosine yeah the see is so here this isn't adjacent over hypotenuse it's hypotenuse over adjacent well coent goes with Tangent that's going to be your adjacent over opposite and the cosecant that's the the reciprocal of our sign so you're the hypotenuse over the opposite well if we apply this idea of a basic right triangle to the unit circle we actually get ways to compute the S the cosine and the tangent and and all these these these reciprocals cosecant secant cent of any point that we have provided we can find the X and the y coordinate you see when we when we apply this if we look at the sign of this particular angle look at your angle tell me what sign is sign should be opposite over hypotenuse right that's what we Define it as over here what's the opposite to my angle what's the opposite to my angle what is it one oh one Why Oh I thought you said one like wait a second we got to review some trigonometry a little bit here opposite okay opposite is y over what sure how much is y over one yeah that's why we get on the unit circle that the y coordinate for that point is sign of that particular angle that's why we get that it's our it's our definition now so sinal y cosine cosine should be adjacent over hypotenuse so the adjacent of our angle is X the hypotenuse well that's one so we're going to get X over one or X that's why on the unit circle when you see a coordinate point x comma y you know cosine comes first and S comes second because we know that the cosine of the angle is the X it's that distance which is how we correlate that to any point so we go oh yeah okay the x coordinate that means the coine of that angle the y coordinate that means the sign of that angle that's kind of neat right it's way we we we combine the idea of a unit circle in our our trigonometry tangent tangent is opposite over adjacent so in our case that's the Y over the X which also leads us to our identity if you look at that what how much is equal to Y over here sign how much is equal to X you can say it out loud it's okay to talk in this class how much is X if we know that tangent is YX y = s and coineal X then this also leads to S over cosine right so we get our first identity tangent equal sign over cosine that's pretty easy to see not too bad now you can also do this with our um with our reciprocals here instead of opposite over hypotenuse as far as our unit circle goes we get cosecant we get secant and we get cotangent they're just the reciprocals of our identities here so if s is y cose is 1 over y if cosine X secant is 1x tangent is YX Cent x y by the way I would like you to do this go go in your book and read not right now go in your book and read through uh I think it's around page 30 through 35 somewhere in there they have a listing of common angles that we use it's going to really be helpful to you if I can say to you uh give me cosine of uh of pi over 3 and you're able to do it give me cosine of of pi/ two I mean at least give me give me sign of pyro 4 things like that it's going to help you a little bit so refresh your memory on that if you need the circle right now for your homework great but I want you to get kind of in the habit of knowing those by heart or by by memory can you guys do that for me at least look them over don't go in this class without knowing at least what I'm talking about okay knowing that those exist and and how to at least find them try to memorize unit CLE in your head got it are you guys familiar with what happens to to S cosine and tangent as we go around the quadrants you guys know what happens with that so for instance uh I know that there are four quadrants to any XY AIS or coordinate system and this is quadrant what was this quadrant and which one's two left or down left good and so that means that's three and this one's four we use Roman numerals because I have no idea we just do so Roman numerals that's how we got 1 2 34 are you familiar with what is positive in which quad Al all wait all students take calculus yeah oh I I always thought it was astrology sucks total crap but I could do anyway no just kidding yeah all students take calculus as a new mod to remember that in the quadrants certain certain trig functions have certain values for instance a means all the trig functions are positive right here all the trig functions are positive in the first quadrant students means sign is positive in the second quadrant what that also means is that cosecant is positive all students take so tangent and third that means cotangent as well and last one is cosine or secant so all students take that's not true though we got to find something that's true we got to find think of a better pneumonic formul like tral TR okay that was original how about uh the acronym surrounding this class I like that yeah that's that's very okay well you come with something better anyway all soon take calculus works just fine tells you what's positive and which quadr now one way that we can use this if you didn't know you're like H why do we even need to know that all students take take calculus or know the values of those quadrants is we can find out the trick function for any angle using idea called reference angles in combination with thec uh knowing the quadrants let me would you like to see how that's done let me show you how that's done so we're going to talk about reference angles just briefly then I'll show you how to combine the idea of reference angles with the idea of this or knowing our quadrants to determine the trig function and its value for any angle that I give you provided you know the unit circle okay that's a must so reference angles uh how to find Tri function any angle is is what this is and the idea is is this what we're going to do is we're going to try to make an acute angle with the xaxis and then we're going to use the find the trig function of that and then use the ASC idea well we really have four cases because there's really only four quadrants so here's what can happen what I'm going to do is give you a picture a representation of our angle and then show you how to find the reference angle let's say we start with just a regular angle right there is the angle acute yeah then we're already done then the angle is our reference angle so right down here the reference angle would just be Thea itself you don't have to do anything with that which means you can find the trig function for the angle pretty easily just by using your unit circle so that would be our reference angle however check it out if I go past Pi / 2 so for instance this one is that angle acute or what I want to do is find the reference angle here's the reference angle it's just the angle that the terminal side makes with the xais which happens to be acute so where this is my angle the reference angle would be this shaded version over here we just need a way to to represent the Shaded version right now so if this is our our terminal side and this is the x-axis how much would this tell me this how much would this whole angle be if it went all the way to here do it in terms of radius this would be Pi sure now would you agree that this shaded angle is pi minus whatever my original angle was that's how we find a reference angle in this case for this quadrant the reference angle would be Pi minus your angle that's going to give you an acute angle that that's formed between the xaxis and your terminal now just you're okay with this so far you see where the pi is coming from PI is coming from because well this is that's a measure of Pi we're subtracting our angle we're going to get that shade and region let's keep going what if we're over here can you see the reference angle I want to make right here this is going this is going past pi and then going a little bit my reference angle has to be an again an acute angle angle between the terminal side and the xais so what I'm looking at is this is the whole thing right that's the whole whole P or whole um angle I want just this little piece so let's see how much is it again from here to here okay now I'm going further this much further would you agree that this angle is all the angle minus Pi look at the angle so this is this is the whole thing right and I want to take away all the non-shaded stuff how much is all the non-shaded stuff that's Pi so our reference angle here would be Theta minus Pi okay last one it looks like packman a little bit in our case our reference angle is going to be this shaded section can you can you think about what that's going to be what do you think in order to find the Shaded section we're we're probably not going to do well we could do well no we can't we probably not do anything with pi but maybe with 2 pi 2 pi would be the Whole Thing 2 Pius 2 Pius the might work cuz 2 pi would be this whole stuff I want to subtract off the the Theta part that's going to leave me with that that cutout nuder if you're okay with that yes yes yes I'll take it so reference angle here okay raise your hand okay on where the reference angle idea comes from you feel all right with it good good now would you like to see how to actually use it yeah yeah probably that might be nice right let's see about that so let's say I wanted you to find you I'll move over here more say I wanted you to find s cosine tangent secant cant cent of 5i 3 can you do it sure if you have a unit circle handy absolutely if you don't have a unit circle handy all you would need to do is me Mize s cosine tangent of your first quadrant then usec that that's what I'm showing you here how to do if you have un Circle it's really not a problem you just go over right and find that stuff but check this out if we want to find sin cosine tangent of 5 pi over 3 using reference angles here's how we can do it the first thing you're going to do you're going to have to locate it which quadrant it's in so in other words you're going to have to grab it so right now as I'm writing right down the next step I want you to graph 5 pi over 3 okay we just practice that on your own paper graph 5 the next step after that we're going to use this idea of reference angles find your reference angle so 5 pi over 3 I'm thinking I got got to cut every Pi into three parts we've already done that today I know 5 pi over 3 is is positive so we're going to be going counterclockwise I'm going to count five pi over 3s so that means I'm going to go 1 2 3 4 so I should be ending at that one did you did that one also okay I'm going to race this stuff just to make it not so bad for us right now I want you to identify your reference angle can you shade it in just slightly shade in your reference angle be careful on your shading we want something between the terminal side and the x axis not the y- axis don't care about the y- axis I want something between the termal side in the xais so do I shade this do I shade this yeah that's what that's my reference angle we're actually in this situation right now so how can I find the value of my reference angle what would I do would I do the pi minus Theta or Theta minus Pi or the 2 pi which one am I going to do sure I know that this is my angle my 5 pi over 3 what I want to do is take the 2 pi and subtract my 5 3 that's going to give me this range so the reference angle would be 2 pi - 5i 3 if you do it right your reference angle should be something that's in the first quadrant it should be represented in that so uh how much is 2 pi - 5i over 3 yeah that's it you have 6 Pi 3 - 5 piun 3 that's pi over 3 are you guys okay with the fraction work so we've located the quadrant we got that down we know it's in the fourth quadrant we were able to use that to find the reference Angle now the idea is I want you to find all your trick functions of the reference angle so find the treat functions of the reference angle so what I want from you is s cosine and tangent of pi over 3 if we know the top of your head how much is a s of pi over 3 do you know yeah that's exactly right off the top of your head no okay somebody else let's all play along here come on let's do uh cosine of pi over 3 yeah cosine pi over 3 is 12 if you have cosine and you have S you should be able to find tangent because tangent we we just found this out is just s over cosine so if I divide s over cosine I'm going to reciprocate and multiply how much is T of p 3 please perfect I'm not going to do the rest of them but you can see that you could do the same exact thing with cosecant secant and cotangent yes would your feel okay with this so far getting those because this is a basic trigonometry here's how the reference angle idea worked with the ascc if you know what quadrant this is in you knew that it was everything's positive here s's positive here tangents positive here cosine is positive there and everything else is negative here here's the deal once you have this this is the reference angle right all we got to do is interpret what quadrant it was in use the appropriate ASC and then we're going to be able to find out s of 5 pi3 and cosine 5i 3 and tangent so if s of 3 is < tk32 s of 5 pi over 3 without doing any more work is s let's see is s positive or negative in this quadrant definitely negative why well because the only thing that's positive is cosine the rest of them are negative you follow me on that so here we go okay what's what's s of 5 3 then it's definitely negative because sign has to be negative there have you already found out the reference angle measurement just use that measurement with the appropriate sign root3 over two I think I've lost some people because some of you look confused you're like I have no idea what's going on here let's see are you okay with STC that's cuz someone beat it into your head a long time ago you just remember are you okay with the reference angles how to find those okay good you're just subtracting or uh from something or you're subtracting two quantities to make a an acute angle with the x-axis are you okay that this angle goes to here and at this reference angle you have to take 2 pi minus the 5i 3 and what you end up getting is pi over 3 are you okay with that are you okay with getting a pi over 3 are you sure I need head NDS or something you can't just look at me okay yes or no that's okay too if you're no yes okay can you find S cosine and tangent of Pi 3 yall should be going yes because that's in the necessity to even be in this class you're supposed to be able to do that right you have to be able to do that here's the only thing I'm asking you to do what I'm asking you to do is look at what quadrant that reference angle's in what what quadrant your angle is in whatever quadrant you're in that's telling you what's positive and what's not you just said you got ASC right now put it to practice this is useless if you can't do anything with it so you know that if my angle was here everything would be positive I would be done cuz my angle would be pi over 3 if if your angle's here you know that the only thing that's positive is sign everything else is going to be negative do you get it if your angle's here you know that tangent is going to be positive everything else is negative because s over cosine negative negative give you positive that's why tangent is positive if you're angles here you know the only thing that's positive is cosine the rest of them are negative do you get me you've already done the leg work you already know the values this was the hard part now you just got to put uh the signs on top of them so we're we're right here we know the only thing that is positive is which one's positive for this cosine positive for this this one I'm going to leave alone is sign positive in this quadrant I'm going to make it negative is tangent positive in this quadrant no I'm going to make it negative and that's all I'm doing right here so I can guarantee you my answers are going to be < tk3 over2 positive 12 and < tk3 that's it s of 5 3 is that because we already did the reference Angle now we know the sign cosine of 5i 3 is 12 why because I already did the leg work I know the value I'm just trying to figure out the sign if cosine still positive there notice if we doing a reference angle it's all based on quadrant run one run run run from this trigonometry stuff if if you're doing a reference angle it's all based on quadrant one that means every number you're going to get is positive that's why you have to usec to find out whether or not it's actually NE do you see the difference there that's the point so here I know coine cosine remains positive I know and tangent do not so in order to get from the pi over 3 to the 5 pi over 3 it's not I'm not Harry Potter I'm not the math version of that right this isn't Magic it's just using the fact that use a reference angle and now you know that if it's in a certain quadrant What's positive and what's not does it make more sense now to you yeah good that was less anticlimactic than I wanted I wanted like like a woo I get it can we try that once no I'm just kidding you don't have to do wo I get it but hopefully the energy level will go up for you someday how were you when you took your class I don't remember I took math 4C in summer school so that that was literally a six week blur I do not remember hard any of it I had to go relearn it awesome don't don't do that to yourself if you're going to do anything with mathematics if you don't if you don't give a crap about this sure take it during summer school but if you actually want to use it and going to have to use it don't take it during summer school oh my God unless you're like a superar math some of you are do it fine I wasn't okay last thing we're going to talk about today in the last about 3 minutes is a little bit of graphing where the stuff comes from uh what I need you to do is read through or at least be familiar with your basic Tri graphs okay I need you to know what s looks like what cosine looks like what tangent looks like s goes like this and cosine goes like this and tangent goes like this okay and know where they start what their period is things like that you need to know that at least if you don't know that you won't be successful with this graphing part so we're going to graph a couple different types of functions here we're things of the form y = a * s of some angle or cosine of some angle and then I'm going to make it a little bit more complicated than that okay we're going to graph another form probably next time where we're going to make this a littleit more advanced so for instance our example for this I'm going to graph y = sinx just our our standard S curve and Y = let's see uh two s of 4X and we're going to see the difference between these two graphs firstly let's talk about sign right off the bat where does sign start or that that one does yeah how high does sign go what's the period for sign Pi that means it oscillates every 2 pi why well it's it's on a circle right it's going to come back to the same thing every time you go around that Circle that means something important is also going to happen at Pi that means something important is probably going to happen at Pi / 2 and something important is probably to happen at 3 Pi / 2 tell me if sign starts at zero what what's s of pi oh boy go refresh your memory on that what was everyone be nailing that what's sign of Pi you're just copying everyone else yeah you're lucky it's zero what's s of 2 pi yeah of course because it's going back to the same thing as Z so s of 0 is 0 S 2 pi has to be zero what's s of Pi 2 tell me that mhm and uh s of 3/ 2 lucky guess just kidding so our function has this nice script this is where we get it from like that what we're going to find out next time is what this number does and what that number does and how it changes it does it stretch it out does it compress it does it make the amplitude go different we're going to find all that out next time then we'll talk about this in combination some translations uh that will about our day and we'll we'll go on to the the last section of our review and finally get to some calculus stuff you excited I'm excited I'm excited well happy Monday uh we're going to go ahead and continue talking about what we were talking about which is graphing some trig functions now the basic ones I've asked you to read through that you should know what s and cosine and tangent look like this is our our basic sign graph what we're going to do today is figure out what all these numbers do to these graphs how we can manipulate that so here's the deal whenever we're looking at something of this form like uh a sin BX or a cosine of BX that a and that b those do something the a in front of our sign or our cosine that actually will give you the do you know that's the amplitude that's exactly right so R A actually the the absolute value of our a is going to give us our amplitude and we mean amplitude from the x- AIS so that's why when we graph s of X well there's only a one out there the a is in fact one which means that our amplitude is one up and and one down from the x-axis does that make sense to you what's our B in this case our B gives you the well actually it's the period uh but what's the b in this particular example the B here is four what's the B here one and we know that in our case our our period is actually 2 pi so what we're going to get from our our B is our period so if our B in this case is one and our period happens to be 2 pi then the period is really given by 2 Pi / B that gives us how often we oscillate or how often we come back to repeating the same exact graph is it that specific example or is that that's in general so in in general your amplitude is given by the absolute value of a so in our case let's let's look at this example right here uh so this is our kind of our notes for right now what is our a in our our case for our new example that we're going to do in just a minute two two so our amplitude absolute value of two is two that means it should be going two high and two low from the x- axis are you with me on that okay so for our example I'll write this out again by the way we need a muffler for your for your sneezes as you go just like that shifting gears that' be cool should invent that see money right there red box and mouth Mufflers done just kidding just kidding uh this works the same exact same exact way for a cosine B so those those are the same exact letters those work the same way so let's look at Y = 2 s of 4X and you've already told me that my amplitude my a is the absolute value of two which is two that means we're going to be going up to two and down to -2 that's what the amplitude says here we also need to find the period the period is given by 2 pi over B what's the what was the b in this case again so let's do do that can you simplify 2 pi over 4 two I need you all to be familiar with how to get the amplitude and the period nod your head if you're okay on getting the amplitude and the period in this case good all right amplitude comes from the h just absolute value the the period is given by 2 pi over whatever our B is we simplify that here's what it means it means that this example our the the stuff in the black writing here this is going to oscillate or or repeat every pi/ 2 how you do this if our entire period is 2 pi here what we're saying is our entire period is Pi / 2 here it's going to come back to that point after cycling that's going to be one complete cycle so can you tell me how many cycles I'm going to get in this Range four four Cycles that's four ranges of Pi / 2 four intervals of Pi / 2 does that make sense to you so the period says how often you come back to normal so it's going to go up down and back up then up down and back up four times in this span what did this essentially do to it did this stretch it out or did that compress it compress very much so compressed it so what this did is it compressed it and it stretched it vertically that's what the amplitude did so here's how you can figure out how to graph it though it's going to start the same spot so we're going to be at 0 0 we're going to go up down and back up what happens with these trig functions like s and cosine is the interesting stuff happens at the midpoints so if we're coming back to normal at pi/ 2 something interesting is going to happen at pi over 4 that's midway between there does that make sense to you so pi over 4 something interesting is going to happen okay and that that means I know this is really small it's hard for you guys to see it so this is p 4 right there power four how about midway between 0 and pi over 4 how much is that 8 so let's cut that in half something interesting is going to happen at pi over8 and 3 pi over 8 that's between here over 8 and 3 over 8 now here's how you go the rest of the way we know that the the original function crossed midway between the r two end points here so midway between at this pi over 4 that's where we're going to be crossing the x-axis again can you tell me where the peak of my sign function is going to take place the peak of my well it's going to be a level of two sure but what's the x value of that do you know all right because we know something interesting is going to happen there in this case the peak was at Pi / 2 if we scrunches all together the peak's now going to be at Pi 8 where's the the lowest Valley going to be where's that going to have take place sure that's that's at this one right here so let me recap for real this time not just a you know recap uh what we do in this case is we find our a we find our B the a is going to give you the amplitude how high and how low you go the B is going to give you a period how often you cycle back to the same exact value and repeat your whole nice function so in our case here we know we're going to cycle within a very small window of Pi / 2 we know we're going to go up to two and down to -2 that's going to be my highest and my lowest point how you figure out the rest of it is just the midpoints of that interval at the midpoint of that interval you know something's got to happen in our case we're going to be crossing the x-axis again at the midpoints of those intervals we're going to be reaching our maximum and our minimum value respectively so then we can graph it's just a normal sign function only we've stretched it this way and compressed it this way now you if you're okay with that okay so we're going to be going like this wow that's like that that's our compression we could fit four of those in that range of space would your r hand feel okay with the soap okay would you like to try one more example to see this before we move on to a translation okay why don't you see if you can do this one I'm giving you0 5x but it could easily be X over 2 or 12x okay so5 12 or X over2 that all means the same exact thing here's what I I would like out of you right now what I want you to do draw yourself a nice looking graph and draw the original cosine function just what cosine looks like normally okay just draw that we're going to erase it but I want to get the picture in your head draw the original cosine function then I want you to find your a your amplitude and I want you to find your period do those three things right now I'm going to do on the board but you should have it on your paper oops for okay so before we start dealing with the amplitude and the period I need you to get the the idea the picture of the function down so the picture of the function just the cosine where does cosine start at it goes where goes up okay goes up so cosine would normally start right here at one and go like that do you have that same graph on your paper right now you sure are you positive so we know we're not going to be starting at zero we're going to be starting at either one or negative one depending on what the sign is in front of our function if we've got a negative sign in front of our function you all should know from just your basic algebra and trigonometry classes that if you got a negative what that does is that reflects it does that make sense so we're going to have a complete flip so whereas this would be the cosine function the negative cosine function would start there does that make sense to you you go up and then back down that's what that one would do okay so we have an idea of what this is going to look like it's going to be some it's still going to be a nice curve like sign was only this time it's not going to start at z0 it's going to start at either a positive or A negative number now have you found your amplitude yet how much is your amplitude three okay how' you find three why isn't it negative3 okay good so absolute value of3 is three and your period did you find your period yet we'd find period by doing 2 pi over whatever our B is in our case our B is .5 if we divide 2 by .5 we're going to get four did you get four Pi do it a calculator if you have to two divided 0.5 is four four Pi have you this so far finding the amplitude and the period okay good all right what that means is that we're going to be cycling from our starting point to our ending point in a period of 4 Pi not not shorter now it's going to be what is that is that a compression or a stretch stretch that's stretch so what you're kind of realizing right now that a number that's bigger than one is going to be a compression a number that's smaller than one for example a decimal that's that's less than one or fraction less than one that's going to be a a stretching out a number bigger than one in front is an amplitude gain uh a number less than one in front is amplitude squishing down okay it's shrinking that amplitude do you guys get the idea all right here absolute value less one so between Zer and yeah absolute value between 0 and one that would probably be the best way to say that right okay good catch now how do you figure out where to start well do we start at three do we start at neg3 what do you think3 why well firstly there's a negative in front of it secondly why don't you plug in zero what's uh what's 05 time 0 what's cosine zero cosine of 0 is one look at you in the circle you got to know these things at least are simple ones if you don't know s cosine tangent of at least zero Pi / 2 pi and 3 pi over 2 and 2 pi you you need to go back do it today do it tonight make sure you know at least those ones I mean those are huge okay you also should know the pi over 3 pi over 4 pi over 6 then you can use reference angles for the rest but you need to know those at least four you with me on that you got it it's got to be tattooed on your brain hurt but do it or on your forehead you read it backwards in the mirror get it down coine Z one right so cosine of 0 would be 1 what's 1 * -3 we're starting AT3 so we are starting here guess what if we're starting there and my period is 4 Pi I'm ending there the interesting things happen to happen in the middle of our intervals that's how these work so if we have a period of 4 PI what's the middle of our 4 Pi here something interesting is going to happen at 2 pi is it going to be Crossing zero a peak or a valley it's only one of those three things Crossing zero remember I've got to I've got to go up peak come down and end here I'll give you a hint it's going to be it's going to be a peak and the reason is the the same thing happens in the middle as it does at ends for instance if you're Crossing zero at the ends you're Crossing zero in the middle are you Crossing zero at the ends then you're not Crossing zero in the middle that's not what that's not what's happening if you're peing at the ends somehow either a value or PE you're going to be peeking in the middle so what's going to happen here is it a peak or a it's not Crossing zero is it a peak or a value a it's a peak do you see why you can't you can't do this okay this is just one repeat of that that that same number so here you're going to have a peak of three that's the only way that can work that's only way you can make one oscillation through that one cycle and come up the same exact the same exact ending spot you can't go up cross here down that would that would do it you can't do it on that uh specific value it's not going to be Crossing zero there what's going to happen at pi and 3 Pi you tell me that that's where you're Crossing at pi and 3 Pi sure remember our original cosine function did that nice swoop just that was that was it we're just making it bigger and longer so it's going to go up and come back down do you guys see where all these points are coming from are you okay with that interesting thing has happened at the middle of your intervals so middle of interal that's 2 pi if you Peak at the ends you're peeking in the middle and you're Crossing at the the other two midpoints of those sub intervals so we'll draw our function not real pretty for me but I'm not an artist so we're going up coming back down that's only way we can complete that graph do you guys have a basic understanding about how to do these cosine and sign functions now how many people do feel okay about it might take some practice right I have to look at those things go back to these notes watch the videos if you want to have you guys all been to the website by the way yeah it's a good place to to find at least a homework and if you struggle on some of the stuff review those videos now there's one one more thing that we haven't talked about we've done stretches and compressions both with amplitude and with the period what haven't we done it's only one other thing we're going to learn how to shift it yeah that's a whole bunch of shift isn't it oh my gosh that's F okay so the amplitude and the period That's not going to change but what we are going to do is we're going to take a look at what happens if it's not just BX what happens if it's like BX minus something it will shift it's going to shift we're going to find out how to do this how to figure out what type of shift it is and uh and then we'll do an example well the first thing I need you to do I'm going to work with um with this one to show you how it's done but this right here this really isn't going to give it to you it's not going to tell you what the actual shift is you got to do a little bit more math with that so here's what we're going to do we're going to force B to be factored out of those two things so here's the idea if we force B to be factored you're going to write it as y equals the a is not going to change the sign you can't change that but when I say you're going to force B to be factored out of it you're going to divide both things by B if I divide both of those terms terms by B divide that by B what do you get good X okay and somebody else divide this by B what do you get netive C over B very good yeah that we're forcing B to be a factor now whether it actually divides it or not who cares uh we're we're making it divide both of those things one's going to be a fraction the other one's going to just be X by itself now you have be okay with that see where it's going okay we're going to do the same thing with with the cosine so I'm not going to show the work again you would just write it the same exact way this C over B that's where we get our translation that's going to be a shift along the x axis so a shift along the x- axis now it's kind of crucial that you get the shift correct got to get that shift right all right make it's fun to say shift isn't it whole lot of shift in this class what' you learn whole bunch of shift letters class whole bunch of oh if you have this if you have minus C over B please write this down correctly if it's minus C over B like uh x minus this amount even though it says minus C over B what that's going to do that's a shift to the right right this is defined as subtracting the the translation so this right here this minus C over B is a shift to the right so if it's minus C over B you're going to shift right if it's plus C over B you're going to shift it left I'll explain to you why right now why this is the way it is in in two different ways firstly algebraically a shift is defined as x minus the shift so minus the translation that's a shift to the right if you have this check it out X plus C over B would technically be x - c over B do you see that it'd be minus the shift that's a left translation right there that's where that negative is coming from so I'm not crazy all right plus I know in your heads you want to go right but in this case that plus means left um how you another way you can think of it is like a timeline okay if you are adding a number to something you are you're um you're speeding up when it happens according to a timeline timelines go this way right with the soonest thing's happening here and later things happening this way if you add something to a number it's not happening here anymore it's happening sooner than that does that make sense to you you're speeding up when that happens you're adding something to it it speeds up when you attain that value that it's an interesting way of thinking about it if you subtract something like this if you subtract something it's slowing it down it's not happening at at year zero it's now happening later it's slowing down when that would happen that's a shift to the right does that make sense to you it's a different way you can think of translations so um we're going to practice this we'll do we only get time for one example with this stuff but let's practice doing all this stuff with a uh we'll do a side doesn't that look fun yeah no not really not really at all well if we look back at the the notes that I just gave you what's the first thing you might want to do yeah you're going to force it to factor who said that good job you're going to force it to factor what number am I trying to divide out in this case the two very good that's our B right we want to factor out the b so go ahead and make it happen I'll write on the board as you write on your paper Factor the B out but three doesn't change I want you to write it like this what should the first thing be yep it's always going to be X right because you're going to factor that b out of there it's still going to be a plus you're not changing the sign at all Now is it going to be Pi Pi / 2 or pi over 4 because you're dividing out another two does that make sense if you do this if you do pi over two which is what you have divided two which is factoring factoring is division what this is is pi/ 4 you reciprocate and multiply does that make sense pull that out reciprocate you're going to get 12 * 2 that's going to be 4 now let's write out everything that's meaningful about this statement right now I want the amplitude so go ahead and do that I want the period to go ahead and do that that's given by your B remember and I want I want you to determine what type of shift this is and by how much so write out the amplitude write out the period and write out the shift just think if you live at home with your parents and you go home and do your homework and they're like hey can you help me this no I'm taking care of some serious shift right now that's be a good line please someone try that bear let me know if it works I always was thinking of those excuses when I was a kid none of them ever worked I wasn't that smart you guys now I'm giving you all the hints okay amplitude how much is your amplitude good yall got that right absolute Val 3 is three the period is 2 pi over the B that's the number that you just factored out so what is our period here yeah you're right I almost wrote right there equals Pi all right the shift now the shift is this going to be a right shift or a left shift what do you think yeah even though it says plus you your brain wants to think it's to the right right but it's not it's to the left this you can think of as it's speeding up it's happening sooner if that's moving to the left according to the timeline or if you think about the definition of what a translation really is it's minus a negative translation so that is definitely to the left of Pi 4 okay well we got that I'm going to show this to you you can graph this with only one graph all right but I'm going to show this to you as the original function this is how I think about it I think about the the original function I think about the Amplified and this is going to be a compressed function and then I deal with with the shift because if you try to do it all at once it gets a little confusing you with me on that so deal with the amplitude and the stretch compression first and then draw draw that one and then shift it so here's what I mean by that the first thing I know it's going up to three and down to3 the next thing I know you you need to know the period how much is the period okay so over here I'm going to put a pi and that's going to be my ending point I know for a fact that interesting things I use interesting things interesting things are going to happen which means points that we're going to are going to be valys or Crossings of the x-axis are going to happen in the middle of this interval so this is my my end goal is going to be this I know I'm positive because I have no negative it's it's going to go down and come back up and end right here does that make sense to you that's what's going to happen the interesting stuff's going to happen here at p over two here at pi over 4 and here at what is that one yeah do you feel okay that we're going to be going up and down to three do you feel okay that we're going to complete a complete cycle in pi time in the in the interval of Pi do you feel okay that this we need those numbers because that's where the interesting stuff is going to happen yes no yes okay so let's figure out what this is remember we haven't done the shift yet we're just going to draw the function without the shift and then we'll deal with the shift we'll just move it over Okie so let's deal with this function if I were to ignore the shift for a second where does it start three it's either three because it's cosine right it's not at zero it's either three org3 in this case which is it definitely you plug in zero you're going to get three remember not not with that though you're ignoring that for second so I would start here that means I'm going to end there got it that means that what happens at pi/ 2 is that a peak a zero or a valley you should be able to tell me right now Valley definitely a valley absolutely you're right here what happens at Pi 4 and 3 Pi 4 then that's our Crossings so my graph should look like oh that's not bad I got that on video too look at that that's pretty feel good good job Le good gra like the best one I've ever done my gosh remember that it's never going to happen again all right this is it would you agree that this is my function without the shift now that shift it said to the right to the left which one left what that means is that all we've got to do is take every key Point boom boom boom boom boom all five of those key points shift them pi over 4 to the left and then redraw it that's all that means so fortunately for us we have almost all the P fours listed right that's cool we just got to make sure that this distance that's p 4 you said to the left so here's what it does it's the same level just shifted oh boy I saw you're confused looks dang it that's my third mistake today I'm passed my quota by one for the week I make two mistakes a week that was it dang it okay I know it's Monday right that's that'll be my excuse too m h yes very good so what we're going to do is take each of these key points we're going to shift them one interval of Pi 4 to the left redraw so they at the same level though so this point was at 0 now it's attive 4 3 that's right there this point was at Pi 4 0 now it's at 0 0 are you with me on how we're doing this right now this was at pi over 23 now it's at Pi 4 that's 43 okay cool this was at 3 pi over 4 we're going to shift it Pi 4 to the left that's now at Pi 2 this point was at Pi Pi comma 3 now it's going to be shift Pi 4 to the left that's 3 Pi 4A 3 we've just shifted every major point to the left that's because we have shift to the left of Pi 4 if you redraw it that'll be the purple line right now see I missed told again we get the actual function this was the one we copied you can't do that if you use pencil uh but this is this is our shifted over version the purple one's the one we need to end with would your raise your hand you okay with our our shift good all right that's good did you think they were going to be as as nice as this it's nice huh do it this way if You' never done this before this is a good way to do it now what I do need you to do is you need to review your trig identities in the book I I think it's somewhere on page 36 or 37 review those things we're not going to be dealing with anything super super duper hard like some obscure ones but the basic ones you absolutely must have like uh tangent equals sign over cosine you got to have that you got to have the Pythagorean one down where sin square plus cosine squal 1 and the derivatives of not derivatives that's a wrong word to use in Calculus class but the U the cories from that such as 1 minus sin squal cosine squ you need those things okay you also need like your half angle and your double angle formulas you need the ones that say uh sine of 2x is the same thing it's 2 sinx cosine X you need those okay so so review those have them in your head somewhere be able to use those things because we will use them when we're doing integrals and derivatives of trig functions you need them