 So here we go. Here is our quiz Cirquee in which quadrant does the terminal arm of an angle measuring 890 degrees lie? I better do a sketch Let's see How far Spencer and then I told you that the trick is it's your nine times table with an extra zero So 360 I think 36 45 54 663 Good morning 72 81 Now right to here would be nine hundred. Oh a little bit too far 890 Right there, which was quadrant Quadrant two. Oh, I could even tell you the reference angle if they asked the reference angle is ten degrees Example two. What's the smallest positive coterminal angle? With negative 170 by the way another word for smallest positive coterminal angle Brett is the Principle angle the angle between zero and 360 that's our principle. It's like our lowest terms fraction. It's Sorry the same idea Easy way to find it is to go negative 170 Plus 360 and I can probably do it my head 360 negative 170 plus 300 is negative is positive 270 Plus 60 is No, I'm botching that completely mr. Do it first thing in the morning. I better go to the machine Let's go 360 minus 170 be 260 minus 70. Is it is it 190? Woohoo. I did recover 190 degrees now It's 360 degrees if I'm dealing with degrees if I was in radians and there was no degrees symbol there I wouldn't add or subtract multiples of 360 to go once around and find coterminal angles But what I add or subtract instead if I was in radians. What's the same as 360 degrees? To pi speaking of which Example three. What's the reference angle for okay 320 degrees 320 degrees is new right there reference angle is that right there? 40 degrees Reference angle for 7 pi by 3 well as soon as it's a third. I'll call this 3 pi by 3 and 6 pi by 3 pi and 2 pi 7 pi by 3 would be all the way to 6 and then 1 pi by 3 further which is about that much And I'm pretty sure my reference angle is this my reference angle is pi by 3 Negative 7 pi by 6 negative 7 pi by 6 means to go in this direction. Here's negative 6 pi by 6 Here's negative 12 pi by 6 negative 9 pi by 6 negative 3 pi by 6 anyhow negative 7 pi by 6 There's negative 6. I'm gonna go 1 pi by 6 for after. I'm pretty sure the reference angle is positive pi by 6 One more each We said reference angles are really handy for something like number four number four when they're asking me to solve a trig equation Where they want me to find theta there is now two answers if they give me a domain between 0 and 360 or in radians between 0 and 2 pi they have asked me to solve a trig equation So the first thing I'm going to do is a sketch And I'm gonna write the cast rule and they've told me that sine was negative Sine is negative here and here. I now want to find the reference angle the reference angle is going to be the inverse Sine not of negative point seven six six just a point seven six six I never put a negative in to find a reference angle. I'm in degrees So I better make sure my calculator is in degrees. Hey, it is. That's amazing. Must have just taught physics inverse sine of point seven six six and I Get 49.9. I'm pretty sure mr. Dewick had 50 degrees in mind when he made up this question That means that this angle right here and this angle right here are each 50 degrees how big is Theta one theta one is that big right there, which is going to be a hundred and eighty Plus 50 degrees which is 230 degrees. Yes and Theta two is going to be this angle right here, which is 360 degrees take away 50 degrees 310 degrees that correct folks people nodding Okay, if you got that you get two out of two Otherwise, I would give you the way I would mark this if I saw that you had the correct quadrant That would be worth a half mark a half mark for finding a reference angle a half mark for the first angle and a half mark For the second angle and there's your two marks Okay Turn the page Says find the other five trig ratios. They gave me cosecant. So I guess I'm going to be looking for Sine theta Coase theta tan theta secant theta And cotangent theta and you'll notice I wrote the pair side by side making dumb mistakes And they told me cosecant. So I left this one blank and they also told me what quadrant we're in because if I use my cast rule Alex which trig function does cosecant go with? So sine is positive and cotangent which goes with what? Tangent is negative. Where is sine positive? Well sine is positive here and here Where is tangent negative tangent is negative here and here? Where is the overlap? Aha, I'm in this quadrant. I have to be I'm in quadrant two which means that for what it's worth Cosine is going to be negative. So we'll secant tangent will be negative. So we'll cotangent and Sine will be positive Not only that they gave me cosecant as a fraction cosecant goes with what did you say Alex? Sine is what over what in terms of x and y and r folks Y over r. Oh, this is r equals 25 and y equals 24 and I'd like to find x which is going to be negative look at what quadrant I'm in mid-suit has to be negative and I know that oh, yeah x is going to be the square root of r squared minus y squared. It's going to be the square root of Mr. Duke you put a 25 there. How about putting a 24 there? That would have been a little silly 25 squared minus 24 squared 25 squared minus 24 squared equal how about 25 squared minus 24 squared equals Hey, do you get x equals negative 7 and now it's filled in the blanks Sine is y over r Cosine is x over r secant is r over x tangent is y Over x and cotangent is x Over y the way I would mark this there's five trig ratios I'd take a half mark off for each one that was wrong. So if you get four of those wrong you get zero, sorry Now number five is exactly the same as number six Well, yeah six and five are the same question except in number five. I gave you the r and the y as a fraction Here I gave you the x and the y as a point so I know that x is negative three y is negative four Well, I need r. I don't know but oh, yeah, they want cosy can't cosy can't beta is Cosy can't goes with sine cosy can't is r over y It's gonna be something well. It's gonna be negative. There's gonna be a four on the bottom. I also know that x squared plus y squared Equals r squared or if I want to spend so I can get rid of the squared on the r by cluing in that It's gonna be the square root of that. Do you end up with r being five? I think it's the three four five triangle that one I kind of got memorized and I think you end up with That yes negative five over four So I will say this as far as I'm concerned Brett I can give you either any trig ratio or Any point and I can ask you to find any of the six other trig ratios great multiple choice question By the way in exact form means leave it as a fraction Not as a decimal exact value means leave it as a fraction and if there have been square roots in here leave them as square roots Yo four is not positive. It's negative Right there the five is positive, but we don't put it you put the negative on the bottom It's not wrong. It's just bad math matters just like walking around something hanging from your nose It's not wrong, but if you want to have friends you really want to deal with it I look for a negative in a fraction on the top or in the front if you stick it in the bottom odds are I'll miss it and If I do see it, I'll go off this person has bad math matters And I'll think bad thoughts about you Again number seven calculator work evaluate the following to three decimal places. I'm in degrees. I better make sure I'm in degrees Yeah, I am How do I find cosy Kent? Well, that's going to be one. Let's try that again. Mr. One divided by the sign of 84 It's not the reciprocal of the 84 most common mistake By the way, that's your answer the most common mistake. I see Tyler's kids do this Oh, that's the sign of one over 84. You're not flipping the angle. You're flipping the trig function. So it's not that It's 1.006 B Radiance It's going to be one divided by the sign of two Pi by three close bracket close bracket Plus he can't goes with cosine one divided by the cosine of Four pi by three. I just typed it all in in one line. I Think you get negative point eight four five Yeah, am I wrong? Four pi by five. Mr. Duke. You don't get negative As I was saying, you get negative point zero eight one. Yes One mark for each of those now. I told you this first trig test is going to be non-calc Which means these two questions would not be on your first trig test. They will be on your second trig test Well in a mr. Duke How could you questions like this without a calculator? There is a way to find a reference ink if you know a Trig decimal to find certain reference angles will memorize certain ones. That's going to be in the next couple of days actually Anyways, can you give yourself a score out of count them 12 please? Making sure your name is on it and passing them forwards, please Can you all open your books to the homework from last day, please which was? Oh, let's put little answers here by the way Which was an introduction to radians We didn't do a lesson last class because of the pep rally in the shortened classes. I Said it was your time to get totally caught up. I hope you did Because for today in the next class, it's going to be applications of radians like crazy And if you don't understand radians, you're not going to have much success in getting the applications So I'll begin any questions from the homework anywhere. This is your chance to throw them at me at least metaphorically not physically yes 7c love to okay Draw the angle and standard because negative 2 pi by 3 this one Draw the angle in standard position first of all negative means I'm going to go that way And I'm going to call this negative 3 pi by 3 negative 2 pi by 3 if this is if three thirds is half a circle I think two thirds It's about there. I'm kind of in my mind dividing that bottom half of the circle into three equal chunks Right. There's one third right there Two thirds three third roughly The reference angle is that right there? Oh, they didn't ask for the reference angle, but I would pi by 3 the principal angle now the principal angle is going to go this way Just the positive angle So first question if I went to here how far am I I think I'm 3 pi by 3 and 1 pi by 3 further I think the principal angle is 4 pi by 3s. It's 1 pi by 3 past 3 pi by 3s One positive and one negative coterminal angle. Okay If I add 2 pi except I'm not going to add 2 pi I'm going to add 6 pi by 3 which is 2 pi, but why not go to a common denominator right away I think 1 would be 10 pi by 3 And if they want me to go negative I'll start with the one that they gave me negative 2 pi by 3 and I'll subtract 2 pi Which is 6 pi by 3 negative 2 take away 6 more negative 8 pi by 3 this this This and this all the same angle in that. I'm sorry all the same terminal arm. You'll end up in the same spot I can keep going ad nauseam as long as you want me to That common denominator trick is a good one use it as soon as they give you whatever denominator they give you start doing your labeling your quadrants in those 6th is easier 3 thirds is a bit trickier because this is a kind of a half and a half of a third It doesn't work out evenly, but you can kind of imagine this divided into three chunks and the top divided into three chunks That's how you get six chunks of pi by threes Is that all right Next So we understand we're good on radians If not I'm around today after school. I can help you one on one You do not you do not you do not want to try and put this one off and think oh catch it Let me not a trip. Please trust me not not your best decision Let us please then go to lesson five applications of radians page 258 And the first application of radians is now i'm going to ask you to solve trick equations in radians Not in degrees So it says example one find each value of theta Then it says this now. I got to be a little bit fussy This has been zero and two pi. That's the same as between zero and 360 The only thing bret. I don't like is for some reason in this book They go from touching zero To touching zero again on a test I will almost always not put an or equal to right there because I would rather have you go all the way around and stop Not all the way around and then get back to where you started from and so you're repeating one single angle I don't like the fact that they're repeating zero degrees So if you ever it's going to be important if your answer is exactly zero or 360 But I won't be too fussy. I'm whether you give me both or not. I just don't like that your wake Good job. By the way, congratulations senior boys won the airshow on the weekend played quite splendidly Mitsu gave some six foot three kid nightmare for the rest of his life when he blocked his shot on a breakaway in front of the whole school You know, he's going to be in therapy some Anyways, okay number one Signs theta equals a half You know, I'm going to do first to solve this to find both angles Physics 12 kids. I'm going to say what does that mean? I'm going to sketch drop picture. So I'll do that over here Write my cast rule Emily which trig function did they give me? Did they tell me sign was negative or positive? Which means I'm uh here and here yes Now I need to find the reference angle the reference angle Is going to be the inverse sign of All right point five instead of a half because it's easier to take one half I could just see I lost a few of you fine reference angle of one half All of you right now put your calculators and radians if you haven't already We did degrees on the quiz. Yes. I think that'll be the last you see of degrees for the rest of this unit Unless I ask you to convert from degrees to radians right but otherwise But your calculator work is going to be in radians. So I'm going to go a second function sign Of one half And I get point five two three Six Now that means that this angle and this angle are each point five two three six And this is where I said The only disadvantage with radians is doing the arithmetic in your head Is a bit trickier Everything else radians far surpasses degrees And so we talked about leaving degrees behind not responding to the emails not looking up the texts not answering the phone Okay, don't open that package of chocolates that's sitting on your back porch It's done. It's over How big is this first angle? Okay, the first angle is point five two three six How can I calculate how big this second angle is? Not 180. Oh, you call them back Brett. How could you do that? He's not worth it. I'm telling you Oh, but he was so nice for all those years. I'm telling you you've upgraded. It's not 180. What is this really? Not 180. What is it? Okay, let's remind ourselves. It's pi So it's not going to be 180 minus it's going to be pi Minus point five two three six, which is what you got the pi button on your calculator. This is why I said Brett doing these in your head a little bit trickier, but whatever. What is three point one four? I use the pi button pi minus point five two three six or you can even go pi minus answer button, which would be really clever Kara, what'd you get to four different places? I heard two points. Sorry Round it off properly 6089 or 6090 2.6090 Yes No People are shaking their heads Pi minus that right 2.617999 Use your pi button. Don't use 3.14 at 2.6180 Yes secant Eric, how do I solve secant? It's a trick question. I don't Secant goes with which trig function Eric Come see me after school seriously Okay I got to grill you then Secant goes with which trig function Justin sign Cosine cosine cosine, so I'm going to flip both sides. I can turn this into a cosine But I have to flip this side by putting it Underneath the one And you'll notice Alex up with the fraction in the front not on the bottom because I have good math manners I'm a classy mathy now what sketch It's being a which trig function did they give me Cosine see technically. Yeah, not actually see cosine. Did they tell me the answer was negative or positive? Was cosine Here and here. Yes. Yes. Yes. Yes. Yes Now let's find the reference angle the reference angle is going to be The inverse cos of one over root two not negative root two just I used the negative already I can type that into my calculator as is Second function cosine of one over root two I get point seven eight five four point seven eight five four which means that that angle and that angle Are each point seven eight five four radians There's theta one. How can I figure out how big theta one is? Not 180 Minus, but what Brett? It's going to be high minus point seven eight five four I get two point three five six two Here's theta two How can I figure out how big theta two is? I heard it high plus Backed second function enter Change the minus to a plus Three point two six No three point nine two seven zero Units there are none radians. We don't write units if there's no units. It's in radians yo We always use inverse to find a reference angle Right, we know the actual Value that they that the trig function worked out to apparently it worked out to exactly root two Which is one point four negative root two which is negative one point four once something or anyways That's the number that it worked out to now. You're going to find starting Wednesday That this number is not a coincidence. There's a certain angles nice angles And their trig functions are all roots and we're going to memorize some of them. Okay What kind of an equation is c it's a quadratic trig? How do I know it's got a squared and a trig It's going to be four answers First thing I need to do is get rid of the squared Now, how do I get rid of a squared? I'm going to square root both sides I'm going to start writing up here because we're going to need some room I'm going to rewrite this as tan theta Equals the square root of three. Oh, but when I square root both sides, I better remember plus or minus I'm going to do my sketch Isabel which trig function did they give me? Tangent was my answer negative or positive turns out isabel both Tangent was positive which is there and there But tangent was also negative because it's plus or minus which was there or there and that's why I'm going to get four answers It's a kind of like a quadratic quadratic Let's find the reference angle The reference angle Brett is going to be the inverse tan Of root three the reference angle is going to be the inverse tan of The square root of three Whatever the heck that is is a decimal. I don't care 1.0472 Now what that means is that this angle this angle this angle and this angle Are all 1.0472 radians There's theta one How big Yeah, 1.0472 How can I figure out how big theta two is? Spencer I just heard you type pi minus 1.0472 which is 2.0944 Theta three is going to be that big Which I think is going to be Pi plus 1.0472 4.18 8 8 Yeah There's the last angle How can I figure out how big that guy is? Yeah Yeah Okay, so we can any quadrant Ryan we're fine. It's either going to be pi plus pi minus or 2 pi minus Or if you're in quadrant one, that's the easy one. The reference angle is the answer Theta four is going to be 2 pi minus 1.0472 Now in your homework Ryan if you don't write this step out if you go straight to your calculator just write the answer I'm good with that in our notes. I'm doing that so that when you're studying You know what the heck we've done just in case you forget, right? 2 pi minus 1.0472 5.23 6 0 Sometimes Just for giggles We can change the domain. We're going to change example 2a just a little bit It says sine x equals that but I want to change the domain. I'm going to change the domain to Let's see zero to pi by 2 zero to pi by 2 Instead of between zero and 2 pi, which means we're going all the way around the circle They're saying finding the answers between here and here. Okay Well, we're going to start the same way Trevor which trig function did they give us? Positive or negative Here and here. Oh, but wait Look at the domain restriction. They said we want the answer between zero and pi by 2 What angle is right here in radians? Pi So what angle is right here in radians half as far or pi by 2? You know what they don't want us to use that They only want us to find this one Okay, I'll humor them I'm still going to find the reference angle Which is the inverse sign of 0.425 However, the reference angle is also my answer because I'm in quadrant one So what is the inverse sign of 0.425? Let's see what you get 0.4. Is that right 0.4390? No, yes I'm not getting any nods Oh, yes 0.4390 Units radians you can write rad if you want to if you have the inner surfer But meanwhile, I just blank is no units is radians This is what we would do Kara if because there's two answers for almost always if we wanted specifically To force you to give us one of the if we were interested in one of the answers We can just restrict the domain like ah, ooh In b Here they want us to go between zero and two pi here. They want the top half of the circle answer That's between zero and two pi. Let's try b Uh Still going to start up by doing a sketch But the domain that they gave us here Eric They want the answer x to be between zero and pi That means they're saying, you know what ignore the bottom just use the top half of the circle. Okay Dominique which trig function did they give me? Tangent is negative in which of those two quadrants Because it is negative. Yes, and now it's going to be the same old same old But we're only going to finding one so reference angle Is going to be the whoop change colors for some strange reason. Mr. Good. No silly It's going to be the inverse tan of four over five The reference angle ends up being Inverse tan of four five point six seven four seven theta one is going to be How could I find theta one Pi minus you're going to go straight to my calculator Pi minus and I'll even use my answer button since it's there 2.4669 Oh, they said to the nearest hundredth, didn't they? Why don't I follow their instruction? Well, whatever 2.47 I guess if I follow the instructions in yours hundredth, and I guess this should be 0.44 Put the self Read example three Find the value of cosecant. Eric cosecant goes with which trig function No, those the the cants can't hey, hey, hey, here's the cants can't go together Hey, huh? Hey Oh, come on. That's pretty good So cosecant can't go with secant. What does cosecant go with? There's a sign So they want me to find cosecant If tangent is that oh and we're in that quadrant there and we're between okay Let's do a little sketch This is pi This is 2 pi This is pi by 2 This is 3 pi by 2 Asar, what's the domain restriction they gave me? Can you read it out? We're between what? Look, ah, no Not 180 We're not we're not we're not calling him again leave asar. I'm telling you you can do better. I know you can't It's not worth going back to Just read it to me between what? and You know what? I think they're saying we're in there c a s t Tyler which trig function do they want me to find? Will cosecant be negative or positive in this quadrant here? Understand you said negative. Yes, this is multiple choice. How much you want to bet you cross out half the wrong answers? In fact, I'll tell you half will be positive and half will be negative because when I make these up That's one of the things I'll do cosecant is What over what in terms of x and y and r? What did they give me they gave me tangent tangent is what over what in terms of x and y and r? So according to this the y value is 1 and the x value is root 3 Everybody see how come we're wrong we are That the y value is not 1 the x value is not root 3 You see it look what quadrant they told us we're in Can y possibly be positive in that quadrant? Can x possibly be positive now tangent was positive because actually it was a negative here And a negative here and alex when you have two negatives in a fraction what can't yeah, you know what it's that and it's that Which by the way is where this is going to be get it's negative from it's going to get it from a negative y We need to find r r is going to be the square root of x squared plus y squared Sand Holy, what's negative squared It's root 3 squared just plain old 3 plus What's negative 1 squared in fact you get the square root of 3 plus 1. What is 3 plus 1? square root Plus or minus no no radius is always positive. You know what you get that Although would I put the answer over 1 if I gave this to you? No now I was question. I like this question. In fact, I'll even tell you what the answers would be This would be a multiple choice question on your test The answers would be positive 2 negative 2 positive 1 half and negative 1 half Because I would assume some people would get cosy can't and forget to do the reciprocal So even though the answer is ahead of time Okay, so there's some applications of radians one more and we're done Turn the page The arc length formula The arc length formula In the last lesson We defined radians the angle. We said it was the arc length divided by the radius theta was a over r Physics 12 get the a by itself How would you get the a by itself? What would you do with this r? Times it to the other side. In fact, this Is the arc length formula and you need to know it Now I'm going to say you need to memorize it But I'd like to show you the easy way to memorize it I've never seen this in a textbook. This is mr. Dewick's invention and he's proud of this. So look up This is the arc length formula Brett the wet length formula Doesn't it sort of look like the word arc? The arc length formula. It's the word arc sort of not quite But that that's how I've always remembered it when everyone someone says arc lengths. I go. Oh a equals r Cut out theta. I got to turn the c into theta This is the arc length formula and you can use this to find an arc length. Oh, but here is the key Theta has to be in radians as our what does theta have to be in? No more degrees That means if they give me an angle in degrees, I'm going to have to convert it into radians And r is the length of the radius Here are some questions we can quickly do A pendulum 30 centimeters long swings swings through an arc of 45 centimeters through what angle does the pendulum swing? Answer in degrees and in radians instead of your tenth. We'll do this in radians and then we'll convert our answer to degrees Do they mention the word arc? Yes Do they mention an angle? Yes You know what I'm pretty sure this is a job For the arc length equation, but what do they ask me to find in this question dominique? Let's get theta by itself. How would I get theta by itself? Theta is going to be a over r What's the arc length mentioned in this question? read yeah 45 What's r? I guess it's the length of the pendulum by the way For those of you who are going what's a pendulum? a pendulum thing that swings Either on a string or this could be a piece of wood like in a grandfather clock The grandfather clocks have a pendulum that ticks back and forth. It can be solid anyways Can you see the bottom is tracing out a circle an arc? Which means i'm pretty sure the pendulum length is the radius. Yes The angle is 1.5 radians. That's part of the answer. They also wanted this in degrees Okay 1.5 times How will I change this to degrees? Oh I want degrees on top And I want the radians to cancel so the pie goes on the bottom degrees goes on the top 1.5 radians Is going to be 1.5 times 180 divided by pie divided by pie And I get 85.9 degrees Almost a 90 degree angle, but not quite Okay, example five A circle of radius three centimeters contains a central angle of 2.4 radians Calculate the length of the arc New word subtended by traced out by The central angle to the nearest tenth the centimeter Oh, what are they asking me to find in this question here? The arc length this looks like a job for the arc length equation Which strange enough looks an awful lot like the word arc And they're asking me to find the arc length. Which of those variables stands for the arc length ellen Is it by itself right? Oh great. This is oh double check our angle. Is it in radians? Then the straight plug-in chug three Times 2.4 Heck I can do that in my head. I think 7.2 The arc length will be 7.2 centimeters number six Says calculate the arc length. Oh, okay a equals r Theta They didn't give me the radius. What did they give me instead careful? Oh the diameter if the diameter is 9.2. What's the radius? You know what I'm going to be really lazy. I'm going to go like this 9.2 over 2 I'll just because I'm going to have to type it into my calculator no matter what why not do it in one step There's my radius up Joel they gave me the angle, but is this angle in radians? No 150 degrees times Pi over 180 degrees Will the degrees is cancel? Will I be left with radians? You know what I think I can go all On my calculator multiply the top divide by the bottom. It's going to be 9.2 times 150 times Pi divided by Oh, I got two numbers on the bottom Now you got a couple of options Kara you might be clever enough to say hey two times 180 is 360 which it is You can just divide by 360 or bracket two times 180 And I get an arc length of 12 up to the nearest tenth 12.0 meters So Brianna I'm going to ask you an arc length question Brianna I'm either going to ask you to find the arc length Or the radius or the angle I figure all three of those are fine. I'll give you the other two Oh one more A circle with center c and minor arc ab measuring 15.2 centimeters is shown If angle a equals angle b Okay, they want me to find the radius. I can do this for spencer get the r by itself. What am I what's it going to be r equals? r equals Like you said a over theta right Did they give me the arc length? Yep now here They told me that this angle Is pi by six and this angle is pi by six. Well, that's not the angle that I want I want that angle If this was in a saw, I'm sorry. I'm gonna have to bring him up again. I know it's an emotional attachment degrees What does every triangle add to in degrees? 180 so what does every triangle add to radians? pie or What's my denominator right now here emily? Okay, a triangle equals That much Right. Is that still pi? Yes. Yes. Yes I have one pi by six one pi by six. It has to add to six pi by six How big does that angle right there have to be? Which is two pi by three and lowest terms. I don't care. I'm going to be going to my calculator anyways I'd better put that bottom fraction in brackets on my calculator 15.2 Divided by bracket four Pi by six and I get 7.3 centimeters some applications of radians Solving trig equations using radians and the arc length equation now Pause here for a second In your homework, I'm going to be giving you some questions like example one. We are going to be asked to find an answer The problem is when you look in the back of the book, his answers aren't going to be decimals They're going to be exact values in terms of pi. I'll just show you for example So here's the answers. He's going to be saying the answer is pi by six and five pi by six There is a way to do these in your head getting exact values That's next lesson What happened when they first published this book these two lessons were swapped and so you already knew how to do this so When you solve number one You're going to get that answer If you take this answer in the back and just type it into your calculator pi divided by six You can see if you're right Okay, when you solve number one, you're also going to get But not by nine that answer When you look in the back just change that to a decimal to see if you're right. Does that make sense? And then on wednesday you'll learn how you can actually get these In particular all of number one and all of number three. You won't get exact values You'll get decimal answers. Just change the answers in the back to decimals and see if you're right What is your homework number one number two number three? four five Six seven eight so far I've assigned one to eight nine Yeah Skip 10 11 12 Okay, now let's go back because that's too much. Mr. Duke. Let me see here number one Skip b skip E number two Skip a and d three skip d There that's a little better