 homework from last day to C. Okay, so were you here last day? Okay. This says the sign of A equals what? A little alarm bell should go up. I said our cast rule reference angle to quadrant approach doesn't work when I have something that equals plus or minus one or zero or undefined. This here I'm gonna have to fall back on X and Y and R and so I start out by saying which I should have done in the beginning of class and I forgot but I'll do it right now. Sign is what over what in terms of X and Y and R and the fact that you're having to look it up you in trouble and if you are having to look it up you need to write it up or say it to yourself or do whatever you need to tonight you need to come back Wednesday with that being a boring fact that you automatically know sign is what over what in terms of X and Y and R. It's Y over R. What this is really saying is that Y over R equals negative one. What we now need to imagine is how could this fraction work out to one? The only way this could work out to one is if Y was the same value as R but negative. Okay here's my sketch because a fraction to be one it's something over itself right. Here's my circle yes and we call this distance here R the radius as we go around the circle. Right here the X value is the same as R. Right here the Y value is the same as R but positive. Right here the X value is the same as R but negative. Ah but right here the Y value and the R value are the same and the Y value is negative and so I say to you my child. What angle is that? There is a slightly better way to do that but I can't show it to you yet. This method that I just showed you is the most tedious cumbersome method when dealing with plus or minus ones or zeros. Eventually what we're going to do is we're actually going to quickly memorize what the sine graph what the cosine graph looks like and we'll do a quick sketch for what it's worth the sine graph looks like this where that's positive one and that's negative one and that's 180 degrees and that's 360 degrees and right here is when you're negative one high and low and behold that's at 270 degrees that's the way we'll probably choose to do it. Once we have the sine and cosine graphs memorized oh and yes the sine graph looks an awful lot the letter S that's convenient. Don't want to erase that Mr. Derrick. Is that okay? Next. Do you get D okay? Because that's also a weird one. D is going to be a cotangent is X over Y if it's undefined Y is zero whereas Y is zero I think here and here and I get zero and 180 and 360 if we go around again. Any others? Yeah. Six. Number six. Theta is a second quadrant angle. What that tells me Brett is I'm here and if I go C, A, S tells me sine and cosecant are positive everything else is negative and they told me that the tangent was equal to this what they've really done is they've told me that Y is equal to root 3 and that X is equal to 5. Oh except one of them has to be negative. If I'm in this quadrant what's negative my X coordinate or my Y coordinate? This quadrant the quadrant 2 that they told me that this one that I drew the angle in this one what's negative over there X or Y? Absolutely. Some of you look like you're confused like you can't figure out what I'm getting. Look you're to the left of the Y axis I'm pretty sure you're in negative X you're above the X axis I'm pretty sure you're positive Y so I have that. I would also very much like to find that it would be wonderful if there's some kind of an equation that would relate X and Y and R together. Is there which one Sandley? In fact we can even go like this yes it X squared hey that's 25 plus Y squared what's root 3 squared just plain old 3. R is gonna be root 28. Now they probably in the back of the book will write that as root 4 root 7. They'll probably write that as 2 root 7 because it is. Now I can find these. Cosine is what over what in terms of X and Y and R? It's X over R it's gonna be negative 5 all over 2 root 7. Is that exactly what it looks like in the back of the book or have they rationalized the denominator in the answer key? Can someone look? That's my way of saying can someone check? Does it actually say negative 5 all over 2 root 7 or does it say negative 5 root 7 over 14? Negative 5 root 7 over 4 so they've rationalized the denominator what I said to you guys you don't need to do that we're not gonna be fussy on that. Now if you're saying to yourself how do I know if my answer is right? Negative 5 divided by bracket 2 root 7 close bracket close bracket I get that or negative 5 times root 7 divided by 14. Geez do I get the same answer? It's the same that's how you can check quickly. If it was multiple choice this is the answer I would have for you to pick from. Cosicant okay. Cosicant goes with which trig function Brett? So it's R over Y it's gonna be 2 root 7 over root 3. Now I'm okay with that I'm willing to bet in the back of the book they multiply the top and the bottom by root 3 over root 3 they probably have 2 root 21 over 3 is that correct? If you can just keep your finger there because I'm gonna be asking that same question on the next one too. Yeah and cotangent that's a Y over X no no no mr. do it that's tangent cotangent that's X over Y which is gonna be negative 5 over root 3 and I'm betting in the back of the book they wrote that as negative 5 root 3 over 3. Is that right? Is that what you're wondering why your answers were different from the back or were you able to get like were you able to get these answers okay? Okay so sorry Alberta. Any others? Well then I have I gotta take home quiz for you. I take that back I'm gonna give the quiz out on Wednesday. So next lesson. Radiant measure lesson 4 which is page 249 today is the day the momentous day we are going to leave degrees behind we are moving on we've broken up with degrees and we are in a permanent long-term relationship a stable relationship with radians. Now degrees is gonna write us letters degrees is gonna send us emails we'll get late night phone calls begging us to come back resist the temptation and what I really mean by that Ryan is every year I have one or two kids who are so uncomfortable with radians they change it all to degrees they solve the question and degrees and then they change their answer back to radians don't do that you will never master it if you do that it's gonna be confusing for a couple of days please trust me if you are willing to butch through if you're willing to force yourself if you're willing to push yourself suddenly it'll be there but if you constantly make the conversions it'll never be there starting out war number one what's the formula for the circumference of a circle of radius R well symbol for circumference is that what is the circumference of a circle what's the equation pi r squared is area what's the circumference they be known off top their head 2 pi r in fact if they wanted me to find the circumference of this circle I could say c equals 2 times pi times 5 and I told you that your test was going to be non-calc what we're gonna do is if there's a pi in the equation we're just gonna leave our answer as an exact value in terms of pi because pi is a repeating decimal it goes on forever yes I know 10 times pi would be 31.415926535897 forget it we're just gonna leave our answers in terms of pi in all of our previous angle make angular measure we've used degrees there's 360 Eric degrees in one circle or one degree is one 360 of a revolution in order to simplify some of the calculations used in trig and calc mathematicians use an alternative angular measure radians radians is better in every single way except one Joel it's really tough to do it in your head that's where degrees is nicer if I say to you what's 50 degrees take away 30 degrees Spencer what's the answer 50 20 degrees what's pi by 3 radians take away pi by 7 radians okay but in every other way it's superior so bear with me ratio radians are defined they're a ratio and it's defined as the length of the arc subtending the angle divided by the length of the radius say what sounds confusing I know here's how a radian is defined if you take a circle doesn't look like a circle mr. do it yeah I know pretty close and you take the radius and then you draw an arc exactly the same length as the radius I'm eyeballing it right about there this here is one radian symbol for radian is RAD or nothing actually technically it's not a unit but we write rad that's kind of a well what it does then it allows us to define an angle in terms of the size of the circle that it's in which is although you don't know it yet Kara going to be very very useful one radian all of you guys go like that one radian is about there if you imagine your arm being the radius and if you move the arc about the length of your arm about that one right give or take okay and the nice thing is whether you're Trevor who's six foot six with long arms or well with shorter arms it doesn't matter you'll go through the same arc length because with Mitsu okay right with Trevor it would still it still work so it says use the diagram and definition to estimate a radian measure it's about that how many degrees it says use diagram to to estimate the degrees not gonna we're leaving degrees behind Emily that relationship is off don't return the calls hang up when you see it on your cell phone next page we are going to have to learn to convert from degrees to radians angles can be measured in either and I am going to ask you to flip-flop going from one to the other usually from degrees to radians once in a while we will go back I've fipped a little bit I've exaggerated a little bit once in a while we'll go back from radians to degrees so we need a conversion factor consider a circle with the radius of our units it says complete the following what's one complete rotation of a circle in degrees 360 what's the arc length for one complete rotation well the arc length for one complete rotation is the circumference of the circle and what was the equation for the circumference of the circle that we looked at about 30 seconds ago what's the arc length going to be 2 I are now stay on this page we said that the measure of an angle in radians is the length of the arc divided by the length of the radius if you want to figure out how many radians in one circle it's the length of the arc divided by the radius what do you notice happens here something cancels what cancels 360 degrees is 2 pi there are 2 pi radians in a circle 360 degrees is 2 pi what do you think 180 degrees is then pi you need to memorize that now you will because we're going to be using it over and over and over and over and over and over and over but much like y over r x over r and y over x for sine and cosine and tangent if you see a quiz or a test approaching and you don't have that at your fingertips you have little chance of passing tells me I've done the homework so one half a rotation in degrees is 180 degrees the arc length for the one half rotation well you know what let's just up to the punch line here is my conversion factor pi equals 180 degrees pi radians is 180 degrees now we have to be a bit fussy in mathematics that's the degrees symbol if there is no unit after the number because technically it's if you look at the units up here it's meters divided by meters or centimeters divided by centimeters radians actually glad technically have no unit so if you see no unit it's radians but because people don't like no unit sometimes they're right next to it no that means actually radians is short for radians in other words that's sine of 90 degrees that's sine of 90 radians you know how I can tell because there's no degrees symbol and I'm gonna be fussy on that from now on if you don't put a degree symbol I will assume you mean radians example one says complete the chart 360 degrees we already said was two pi radians the one that I have memorized is this 180 degrees is pi radians how does that help me well 90 degrees is what fraction of 180 degrees 90 degrees is pi by 2 we you can say pi divided by 2 pi over 2 for some reason mathematicians that picked a bit of a shorthand when it comes to radians and they just say pi by 2 and we clue in that you really met pi divided by the number two oh 60 degrees is what fraction of 180 degrees pi by 3 45 degrees is what fraction of 180 degrees it's one quarter 45 degrees is pi by 4 30 degrees is what fraction of 180 degrees 30 degrees is by six you see why that pi 180 a handy one to memorize you can get it off a lot of them fact you know what we can even do a bit more than that you guys keep that little chart in front of you 120 degrees isn't that 260s yes what was 60 in radians 120 degrees that's 2 pi by 3s it is that I can find any multiple of 30 or 45 or 90 or 60 it's not that hard not that hard 210 degrees hey that's 7 30s what was 30 dummy you know what 270 is 7 pi by 6 okay what about 113.2 degrees mr. okay we're gonna have we're gonna come up with a mathematical algorithm that will let us handle anyone but for a lot of the most common that nice round numbered degree angles it's not that hard to jump back and forth okay says the rule to convert from degrees to radians is to multiply the angle in degrees by hmm let's look at this 113.2 degrees what did I tell you to memorize what was the conversion factor what two angles do we know are the same 180 is the same as okay I know this I know that 180 degrees is the same as pi this is my conversion factor if I want to go from degrees and it's a yucky number to radians I'm gonna multiply by pi over 180 degrees why do I know that I want the 180 degrees on the bottom because if you've done physics or chemistry what we're really saying is the degrees cancel we're using a little bit of unit analysis and then I would get out my calculator for a yucky one like this I would go 113.2 times pi divided by 180 the rule is to multiply by pi over 180 I truthfully don't memorize the rule I know it's either gonna be pi over 180 or 180 over pi and I want to make the units cancel so I end up with the units that I want on top if I wanted to convert 270 degrees now there's two ways to do this one the first way is to say hey that's 9 30s or 3 90s in other words if it's 3 90s it is 3 pi by 2 that's the intuitive way but if I want the algorithmic mathematical way I would go like this I want to get rid of degrees so I want my degrees to be on the bottom the pi is going to be on the top and when I do that I'll get 270 pi over 180 and I'll also write that a little bit neater 270 pi over 180 no calculator put it away I'm gonna suggest that the 10s cancel it did say give you a good answer as an exact value in terms of pi 27 over 18 Dominique what goes into the top and the bottom evenly oh so are you telling me that this is it is 3 pi by 2 which I said because I noticed it was 3 90s either way gets you there Ryan 315 degrees it's a certain number of 45s I think it's 7 45s but because I'm not I think it's 7 pi by 4 but because I'm not sure I'll use my conversion factor I'm gonna multiply this by pi over 180 and I get 315 pi all over 180 the degrees cancel now to reduce 315 over 180 and if this was not on the calculator section of the test and it will be well what number goes into the top and the bottom definitely 5 yes yes but most of you don't know that 45 goes into the top and the bottom it does let's suppose you only saw the 5 what's 180 divided by 5 well let's see if we can figure it out what's 180 divided by 10 18 what's 180 divided by 5 twice as much 36 because 10 is half as much as 36 what's 315 divided by 5 I don't know well 300 divided by 5 is 60 15 divided by 5 is 3 I would get that so far if I was hops watching my way there oh and what goes into 63 and 36 evenly 9 7 4 and hey I was right I said I was pretty sure it was gonna be 7 pi by 4 can be done with or without a calculator example 3 says convert the following from degrees to radians what does it say in brackets Ryan that tells me I'm supposed to use a calculator my little tricks won't work so 70 degrees I'm gonna multiply by pi over 180 degrees so that my degrees cancel and I'm gonna get 70 times pi divided by 180 which is what on your calculator ah where's the pie button mr. do it right there 70 times pi divided by 180 70 degrees is about 1.22 radians up to the nearest 10th 1.2 units you can write rad it's not real wrong but there's no units no units is radians 205 degrees gonna multiply by pi over 180 degrees because I want the degrees to cancel it's gonna be 205 pi over 180 do I need to press times oh I don't need to I press times in the previous one I guess I don't need to a little shortcut 3.6 radians that's going degrees to radians occasionally you'll also Eric need to go from radians to degrees so two pi radians is how many degrees did we say 360 which means pi radians is what 180 half pi pi by 2 well what's half of 180 90 a third of pi pi by 3 well what's one third of 180 60 pi by 4 a quarter of 180 45 pi by 6 oh 30 or pi by 180 what's 180 of 180 one degree is the same as pi over 180 the rule is I guess this time we're gonna multiply by 180 degrees over pi and that will introduce degrees to the equation again but I don't memorize the rule I know the conversion factor pi 180 degrees pi 180 degrees and I just ask which one needs to be on the top which one needs to be on the bottom to cancel out my original units pi by 4 oh that's 45 degrees because it's sitting right there negative 7 pi by 3 well the answer is gonna be negative Rihanna what was 1 pi by 3 what was a single pi by 3 how many degrees 60 what do you think seven of them is whatever 7 times 60 is 420 and Carly wrong right half mark off correct maybe fussy now now those ones were ones we could kind of pilot doing our head I'm also gonna give you yucky decimals how do I know I'm supposed to use my calculator for number six Tyler what is it saying about brackets okay so now I'm actually gonna use the conversion factor it's going to be 1.57 times I want the pi on the bottom I want the 180 degrees on the top so that the units that end up on the top are degrees what is 1.57 times 180 divided by pi should be really close to 90 because 3.14 divided by 2 is 1.57 it should be really close is close to 90 what do you get eighty one nine point nine five and it says to the nearest tenth so if you round it off properly it's gonna be eighty nine point nine nine no it's gonna be it is gonna be ninety yeah three point two if I want to change that how do I know this one is in radians it doesn't say radians no degrees symbol three point two hang on mr. duke I wrote that one wrong I went from radians to dig let's try that again mr. duke times 180 degrees over pi I want the degrees to be on top 3.2 times 180 divided by pi 3.2 radians 183.3 degrees try see on your own negative 1.4 radians how many degrees is that yet negative 80.2 yeah degrees so here is the conversion chart if you're someone who just has to memorize stuff you can memorize that I just know the conversion factor pi is 180 pi is 180 degrees pi is 180 degrees and then Carly I use what I've learned from physics about canceling units and tricks like that I want the units that I want to cancel to be on the bottom I want the units that I want to end up with to be on the top oh you can also use two pi equals 360 can we all agree that pi and 180 are nicer numbers to deal with because there's no two okay example seven use your calculator in degree mode to find the value first thing we need to do is we need to make sure our calculator is in degrees press the mode button which is right there make sure you're in degrees and they want us to find the value to four decimal places so sine of 45 0.7071 0.7071 how do I find secant the answer is I don't don't have a secant button what is secant go with okay secant of 135 is actually going to be one divided by the cosine of 135 the most common mistake here Eric is kids instead of going one divided by the cosine they co cosine of one divided by three 135 they want to flip the degrees and not the it's the reciprocal of the cosine not the reciprocal of the angle the secant of 135 is degrees is negative 1.4142 negative does that make sense oh yeah cast rule it should be negative B says use a calculator in radian mode okay now is the time finally at long last mode radians but I like degrees mr. do trust me degrees was a high maintenance relationship it was gonna end up being one of those where you just went oh if I'd known what I was getting into I never would have committed to this ever radians although it's gonna take a little bit to break the ice and get the conversation going oh I'm telling you'll just be like I love such a great such a great part of my life now sine of pi by 4 sine of pi by 4 by the way it's the same answer because 45 degrees is pi by 4 radians so I should get 0.7071 secant of 3 pi by 4 that's gonna be one divided one divided by the cosine of 3 pi by 4 and 3 pi by 4 is 135 degrees this should give us the same answer it does negative 1.4142 so to me now it's fair game to ask you either degrees or radians can you try all of example see on your own all four questions here's a hint do all of the radians first and then do all of the radians or do all the degrees first and then do all the radians but don't flip for each question little trick I'll freeze the screen and I'll do the answers and then bring them up see if you get what I know let's try that again by the way Brett the very last one 45 radians or degrees okay radians why now there is no provincial this year but they love this question they love to give the familiar angle like 30 or 45 or the really tempting one was 90 and they would not put a degree symbol next to it but they would have the degrees answer to pick from on the multiple choice in fact if they were in a really mean mood Alan would make that answer a so it was the first one you spotted and they would get a lot of kids now we need to get used to the cast rule and reference angles with radians turn the page please says in each of the following one draw the angle theta in standard position to state the principal angle three find one positive and one negative coterminal angle we're gonna add a fourth thing I froze it twice and we're gonna add a fourth thing thank you Spencer the fourth thing is this find the reference angle and here I'm gonna do you a favor when I first started teaching this oh good gosh 12 years ago now I was terrible because I do fractions intuitively you go to fractions most of you are not so I'm gonna give you a very easy one-second workaround that will help you navigate your way through this and we're gonna start out by looking at example a 3 pi by 4 I have no idea where that is here's what we're always gonna start up by doing we're gonna say that's pie because how many degrees is that right there hundred how many radians hi is that okay what number is in front of the pie it's invisible a one what's my denominator that they gave me in this particular fraction degrees or sorry this particular fraction radians instead of writing pi I'm gonna write this 4 pi by 4s is that still one pi right that and then put your pencil down for a second and turn your brain up Trevor how far to here you just swore in my class we've left degrees behind she's not worth it come on Trevor leave her behind that's not what I wrote ready how far read that can you say 4 pi by 4s 4 pi by 4s okay nice and loud again how far how far think about it if all the way is 4 pi by 4s how far let's write that down by the way although it's unnecessary how far 8 pi by 4s yes 6 pi by 4s here's why we went to the fractions what what angle did they give me if that's 2 pi by 4 and that's 4 pi by 4 can you tell me exactly where 3 pi by 4s I think 3 pi by 4 is exactly halfway is it not yes yes yes no Eric yes no yes no yes well have I lost you if I have I can reach yes I've lost you or yes you got it sure because I had no reaction from you whatsoever so the angle is that right there we've drawn it in standard position part 2 said state the principal angle now the principal angle in degrees was always between 0 and 360 what will the principal angle in radians be between it'll be between 0 and 2 pi is this between 0 and 2 pi then it is the principal angle 3 says find one positive and one negative coterminal angle for the angle theta well if we were in degrees and we wanted to find coterminal angles do you remember what we added or subtracted 360 what about for radians I'm gonna add or subtract okay so I'm gonna say 3 pi by 4 plus 2 pi and I'm gonna say 3 pi by 4 minus 2 pi no I'm not that's dumb I'll write that just this once but that's why confuse myself back to our diagram are you ready are you ready are you ready Trevor are you ready scene one act one take two how far how far 8 pi by 4 yes 2 pi is 8 over 4 what did I just say 2 pi is what I'm gonna go common denominators right away from the very start my common denominator is always gonna be the bottom of my radian fraction because what is 3 pi by 4 plus 8 pi by 4 how many pi by 4 11 pi by 4 so and what's 3 pi by 4 take away 8 pi by 4 negative 5 pi by 4 and the last thing I asked for was the reference angle the reference angle is that one right there so we said reference angles were defined as Joel what's this measure right there what's this measure right there what was our angle can you tell me how big this has to be doesn't have to be a single what yes if you don't label them with common denominators this gets way way way tougher way tougher don't write this down just watch we're gonna do one together what if I gave you 11 pi by 6 I would go like this no what's my denominator this is 6 pi by 6 what's this then 12 pi by 6 by the way what's this then 3 pi by 6 what's this then 9 pi by 6 where is 11 pi by 6 can you see Joel it's just shy of 12 pi by 6 that I can tell you exactly how shy I can tell you the reference angle already how much below 12 pi by 6 is 11 pi by 6 what's my reference angle 1 pi by 6 oh and if you want me to find coterminal angles I'll add and subtract multiples of 12 pi by 6 11 plus 12 23 pi by 6 is a is a coterminal angle 11 minus 12 negative 1 pi by 6 is a coterminal some of the denominators aren't quite so nice like thirds but that's okay b pi well actually it says negative so this is going to be negative except what's my denominator here Eric I'm going to call it negative 3 pi by 3 and that makes this negative 6 pi by 3 okay we're going this way um what's this well it's half of 3 pi by 3 what's half of 3 you know what this is and I'm going to write this even though it's sloppy this is negative 1.5 pi by 3's is it not right it's half of 3 of them that okay Kara Kara what angle did they give me what angle did they give me I'm sure did you speak you might have oh we're going to go through this one again are we you're awake now say what angle did they give me negative pi by 3 in fact negative 1 pi by 3 I'm going to think to myself because if this is negative 1.5 pi by 3 I think negative 1 pi by 3 is a little shy of negative 1.5 pi by 3 is it not Trevor the other way to think about it is this from here to here is three thirds there's one third there's two thirds there's three I kind of divide it into three chunks in my head except I'm going to get rid of that second line what not the first line just the second line Emily is that okay or have I lost you you're not convincing me your face is saying I have no idea what's going on your tone is saying I'm confused your voice is saying I got this right now two out of three things are telling me I need to re-explain ask what's your question okay how far negative what how far negative what negative 3 pi by 3s what's half of 3 pi by 3s negative 1.5 pi by 3s right what's half of in other words our our our measure it's going to be pi by 3s 3 of them to get to there what's half of 3 what's half of 3 what what what what so half of 3 pi by 3s would be 1.5 pi by 3s negative because we're going in the negative direction that's just telling us to go that way right is that okay yeah because I can tell you now the principal angle now the principal angle is this way it's that way let's come back to the principal angle in a second I want to do the reference angle the reference angle is this big how big is this heard it the reference angle is pi by 3 how's that help huge why did you do reference angle first and then do print for watch watch watch watch watch watch ready carly look up we're going to go in the positive direction now how far all the way around the circle once and don't tell me to pi we're using a common denominator how far all the way around the circle once no no no we're going this way we're going this way we're going this way we're going this way we're in the positive direction now how far all the way around the circle once positive 6 pi by 3 what was the reference angle so how far to hear you're missing it ready ready guys look up how far all the way around six how big is this one how far 5 pi by 3s yes that's our principal angle I can't hear have it work the angle between zero and two pi the smallest positive angle in degrees it was the angle between zero and 360 because for all of these there's an infinite number of angles that we could use we keep going code terminal code terminal code terminal code terminal yes principal angle we defined as way back when in lesson one before christmas we defined the principal angle as the angle between zero and 360 the smallest positive angle is that okay you sure okay almost done example nine says find the reference angle for the following rotation angles anytime i'm doing radians if they give me an angle i do a sketch five pi by six that's pi that's two pi except what's the denominator that they gave me here isabel i'm going to call this six pi by sixes because that's still one pi what am i going to call two pi 12 pi by sixes how big is right here three pi by sixes because it's halfway to six pi by six how big is right here nine pi by sixes yes yes what angle did they give me okay five pi by six i think isn't it right there just shy of six pi by six can you tell me the reference angle what's left over ah negative five pi by four negative means we're going in this direction so this is negative pi well no it's negative four pi by four which is still negative pi this will be negative eight pi by four which is negative two pi how big would this be right here really how far how far hello how far hello negative four pi by four how big is halfway then how big is this up here spot the pattern what angle did they give me negative two pi by four negative four pi by four negative five pi by four i think is right there because six pi by four negative six pi by four is too far there's my reference angle how big carry you're right loud yes what's my denominator here i'm going to call this instead of pi three pi by three and six pi by three so you're ready here we go here we go here we go how far sorry three pi by three how far six pi by three how far nine pi by threes how far if i went to there 12 pi by three too far how much too far in fact i think i want to go there what's the reference angle you already said it when i asked you how much too far what i was really saying is what's the reference angle we need to practice this a lot okay moving around the circle in radians needs to become fairly casual i won't say second nature but almost homework one two three four i'll skip five six seven eight nine ten eleven so i think i skipped five need to practice this good news is i believe next lesson is a bit shorter with less homework let me double check applications yeah shorter lesson next lesson take home quiz next class physics 12 tutorial today after school