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May 27 - 2nd period

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so there were two even-numbered problems and they're basically the same question so the point that was being hammered home here was frequency is reciprocal of. Obviously you needed to use that at the beginning of the homework assignment when they tell you the frequency wave in hertz you obviously need to know that frequency is be over to pie so that you could solve for b in order to do those problems are there any questions on any of the problems other than 2 or 28 number 15 okay so number 15 guess was the first question asked in my other class I guess people ask about number 15 does the answer in the back of the book was a real head-scratcher I agree to answer the back of the book is pretty dumb however even though it is dumb you should realize if you got this answer that the answer in the back of the book is correct I hope you didn't change your answer I hope it's got this answer which is the answer you should have got I hope you kept it and then decided I'm going to ask about that cuz I think I'm right and back of the book as an unnecessarily complicated answer and you didn't write the answer in the back of the book is unnecessarily complicated however even though the answer in the back of the book is unnecessarily complicated how do you know it's still correct if you think this is the correct answer you should be able to look at the answer in the back of the book and say that's all autocorrect we're both correct incorrect answer the back of the book is correct even though it's kind of a stupid answer but I know it's correct Oquirrh incorrect. He just gave me the answer to my question Jonathan crack page cracked Adam. Is 4 so if you horizontally translate 4 units the. Is 4 the same way in fact you could horizontally translate by any multiple of the. this answer and in parentheses right x - 8 or better yet x + 400 is still correct answer because you'd be translating buy some mulch . when you take a wave which of course is cyclical it's repeats itself and you shift it horizontally by a multiple of a. It's still the same way cuz the way it goes on forever and ever and ever you know be somewhere like horizontally shifting you know horizontal line you know you horizontally s*** a horizontal line it doesn't change still the same horizontal line or any multiple of its. It's still the same way I wrote this answer down just to point out you know that if you wanted to you could have come up with the answer to number 15 in terms of sine I have no idea why you would want to do that though I think all the problems in the book the ones that you could do without regression they're all cooked up so that you never had to do a horizontal shift so number 15 doesn't require horizontal shift cuz you could use negative cosine if you get the grass that starts at its lowest point cosine starts at its highest point so if you use negative cosine that'll flip it over to make it so it starts at its lowest point so you don't have to do a s*** if you insisted on using sign the shift would have to be one unit to the right cosine starts at the origin but this way doesn't this way doesn't start at the origin this way starts one unit over so that means if you're going to do this in terms of sine it'll be like taking the origin moving at one unit to the right moving to the right b - 1 see if any of you guys can prove to me that you understood what I talked about 2 minutes ago what other number could I put here some other number other than one that I could put here and it would still be a correct answer technically Kirin tracks standard correct Kira correct funny answers. Will correct show about a wrong answer is the correct answer to my question we something like 5 if this was -5 David correct or some other than 5 - 9y - 504 - 9 or - 13 is all of those horizontal shifts for Eunice say so if you like make this -5 it's like shifting in an additional 400 that would be like shiy additional. Which is fine it won't change the graph how does a plus I can make this X+ what and it would be correct crack x + 3 x + 3 that would also be correct because x + 3 is 1. Shifted over from X - 1 think about the difference between positive 3 negative one positive 3 negative one is 4 units what are the questions we got here question number 19 number 19 says I got a flywheel the candle at the edge of the flywheel is 19 above the ground the lowest point is 4 feet find Mills 5 ft in diameter okay so this is what it's going to look like can you go from 4 9 and that of course means it has that far foot diameter and where is it start to handle starts at the lowest point and it does one route mission the wheel makes a complete turn into seconds so I would be to so that's it a graph in SATA graphic didn't know but now you just have to come up with the model for this negative cosine I can see that midline is halfway between 9 and 4 so 9 + 4 1313 / 2 is 6.5 so the midline amplitude would be the distance from the midline up to the highest value which is at 9 that would be Ada. Is to 2pi / B is the. So that would mean that be to Pi / Pi would be too so he must be pie so the final answer just got to get my units right here doesn't tell me what does height is the a variable I need to use for height so it starts Lowe's of negative cosine and amplitude is 2.5 and I just figured out that b is pi so that would be cosine i t + 6.5 which is my main line me check my answer when I'm resistant or 19 that's cracked that's correct that's crap again this problem feels identical to the example about a jump rope that I had in my notes I think it was example read my notes exactly the same as that you just got to get your midline by averaging the highest and lowest points distance from the midline to the top is your amplitude so there I got my amplitude never got my midline then I just got to fix be so that it has the correct. 2 pi over b equals 2. Boom there's my baby any other questions on the homework this is consistent with what I said which is I don't think you ever had to do a horizontal shift or figure it out yourself the only horizontal shifts in this homework one day you get when you do regression sinusoidal regression that reminds me there's something someone asked me about number 21 on number 21 when you're all finished you got your answer the B value on number 21 was like 0.549 which means that the. When you did number 21 for Superior formulas to Pi / B. Ends up being two pi divided by 0.549 .44 I only bring this up because I was extremely confused by this I hated that I had to answer the question the way I had to answer the question it says like interpret this. Of course this. Is how long it takes this is in mud how long it takes for the the temperatures to cycle through so it's the length of you know one cycle of temperatures 11.44 months so this is a cycle of temperatures in Apple Valley California why is this traveling you put it that way why should this make you if you and show you that Math teachers in the past I encourage you to think when you do story problems think about the reasonableness of your answer how reasonable your answer is yeah Kara Sophia I mean you should expect this to be 12 months don't temperatures you know depend on you know what time of the year it is and I could just come down to 11.44% Charles Mi whatever model I came up with using regression in a pretty bad model because it's telling me that the. Is closer 11 months and is to 12 months I mean we know that the. Should be 12 months so yeah I mean it's the answer to the question maybe they did this intentionally so that you realize you might be getting the best fit wave in this case but the best fit way for this particular set of data could give you a model which you know is wrong this can't possibly be a good predictive model especially if you want to use this model to predict the average temperature January 2030 if you use this model to come the average temperature you know January 2030 you are going to be very disappointed by its inaccuracy so I guess sometimes you know when you're using a bottle you come up with using regression sometimes have your common sense Trump everything else so if I was going to give this model to someone to you I was fixed be maybe I would keep the other numbers the same but this is not the right value for beat and you know that because you know a little bit about a Stronach write a little bit about how the Earth the seasons go on the correct number for this instead of being Point 5494 B how is Unum presenting this as a good model for values Valley temperatures in the future what would be my value for B I would want Siri to be 12 months so the track right now sorry good B would be 2 Pi / 12 whatever 2 Pi / 12 is that should be my B2 hi / 12 now I can get a good deal would be about 25 to 3/6 5236 instead of .5 for 9 any other questions on the homework nope okay Mana this is the last week of new material only two more sections coming up on the end here so we only have two more sections last two sections have to do with the use of identities set alarm this section is all about simplifying expressions in trigonometry let me give you an example before I jump into how to do these problems what kind of bottle of Paige here this is my first example I don't really want to show the work yet just look at this expression doing calculus on this expression here is time-consuming deliberately being vague and just saying doing calculus things like finding derivatives in the past I may have mentioned the word integration calculus is what it sounds like it is a branch of mathematics you know to vote calculate its importance in calculus that you try to work with simple Expressions because doing chart Nation's on complicated expressions is complicated kind of know. You know right more complicated expression is the more difficult it is going to be to work with so your goal in this section and for a couple of weeks next year when you do trigonometry next year you'll be spending at least a couple of weeks on what I'm going to go over right now how to simplify expressions that have trig functions in them so here's an expression notice how it has two trig functions in it it would be nice if we and we will do that here's another example here's an expression that involves three trig function very complicated right it'll be nice if we could somehow write this as a single trig expression these are all things that you do in a calculus problem before you start doing calculus if you're asked to find the derivative or antiderivative of this expression the smart approach would be to simplify the expression and then find the derivative or antiderivative of that expression so today is all about taking an expression making it simpler and these are the tools each one of these equations you see up here is called an identity k a trig identity is an equation involving trig functions where you say one trig Function One function valving trig function is equal to another you are going to be actually creating your own identities these are given identities and you're going to be creating your own when you save this is equal to this you're creating a trig identity you're saying this is identical to this so you're creating a trig identity when you say this is equal to this your crane the trigger you're saying to trig expressions are equal to each other so you're going to be creating your own by using ones that were already created sample this trig identity tangent is sine over cosine now I gave you this identity back in 9.5 I wrote it and was part of the notes + 9.5 I'm not sure when I was going I told you why it was true I probably did but you know now all that's important for doing tonight's homework is to know that this is trip tangent is sine over cosine and we're going to use that fact to do problems like for example to do this problem we're going to rewrite tangent is sine over cosine to do this so you need to have identities in your toolbox so that you can create your own identities that pretty much all of the identities you see up here arrived at by just looking at this picture I don't expect you to have this picture memorized the number of students stains in the United States of America that are familiar with this diagram is extraordinary small most high school students don't even think about this diagram they just memorize identities memorize tangent is sine over cosine and sine squared plus cosine squared equals one memorize back I think you may have been told this one last year this one right here so you might already have this memorized memorize memorize memorize you knew why these things were true and that's why I provided you with his diagram a week ago like what is first one tangent is sine over cosine I think I told you that this was true when I first showed you this why was tangent equal to sine over cosine when I showed you this diagram a week ago why is tangent sine over cosine triangle similarity correct Anthony because in this diagram tangent is to what as sign is cousin why is cotangent cosine over sine the answer is also triangle similarity cotangent is cosine over sine because the biggest triangle in this picture which is this big monster here is similar to the smallest triangle in this picture why is the biggest triangle in this picture similar to the smallest that tangent is tawan is equal to sine over cosine tangent over one is sine over cosine because the Little Triangle in the medium best triangular obviously similar because not only do they have right angles they also share this angle so it's really obvious that the medium sized triangle Little Triangle are similar 1 as sign is cosine what's less obvious is that the biggest triangle in this picture is similar to the smallest so now I'm asking why is the biggest similar to the smallest is anyone see why the monster triangle is similar to the littlest triangle in this picture why are they similar Anthony correct Annie correct anyone else see it I only got the correct aliviane your answer is a fact it's not the reason okay Jonathan you're on the right track but you're using the wrong tabular it's not called the opposite angle it's not called the opposite Kyle same thing you just using the wrong with cabulary you forgot the vocabulary from last year there call alternate not opposite or alternate I'm glad to see at least four you see it on the lower left-hand corner and this angle in the upper right hand corner or call alternate interior angle those angles are actually congruent to each other cuz there's a z c that is z me have alternate interior angles are like for Mazzy so you have right angles that are the same interior angles that are there for the big triangle and the small triangle are congruent to each other now take a look at how you can see that cotangent is cosine over sine the way you do you say cotangent is to 1 and cosine is to sign how do I know because I already said they're similar to each other now I got to do is get the correspondence of the sides correct if you look at the cotangent side what connects the connects the right angle with betta so the angle in the small triangle that I'm sorry this side in the small triangle that corresponds to cotangent has to be the side that connects the right angle with look at which side connects the right angle with SATA its cosine Soco tangent and cosine are corresponding sides because they connect the right angle it connects the right angle Theta and then of course the other leg which is one in the big triangle and sign in the small the other leg Pastor correspond also cotangent / 1 equals cosine over sine because cotangent correspond to each other and the one and the sign correspond to each other so both of these are true because of triangle similarity the lot of you just know this because you know that coach engines reciprocal of tangent a remember when I first told you that coach engines reciprocal of tangent I told you something else and now it's not something that's made up it's from this so why is sine squared plus cosine squared equal to 1 look at the picture why is sine squared plus cosine squared equal to 1 okay Anthony Road Sagarin theorem Jonathan same thing but what is true about sine and cosine and one why is it appropriate to use the Pythagorean theorem here because sine cosine and one r what enables you use you guys who are typing in the tigers are there sides of a right triangle Bingo answer they are sides of a right triangle sign go to sign in one are sides of a right triangle that's the reason why science is cosine squared Why do you think one plus cotangent squared equals cosecant squared again look at the picture why is one plus cotangent squared equal to cosecant Square or is the answer to this question is again what Anthony type in which has their sides of a right triangle again which right triangle 1 + 13 squared equals cosecant squared has one cotangent and cosecant are sides of a right triangle which right triangle are they sides up yeah the biggest one is size of this monster triangle 1 cotangent has really long I potnis is called finally tangent squared plus y equals secant squared those are all sides of the medium sized triangle squared + 1 squared equals c squared so there are three tickets Zachary and identities there are three pythagorean identities because this truck this diagram has three right triangle for each right triangle in this diet there's another Pythagorean identity because in the diagram that when all the stuff originate from can there are three right triangles that's why there's three pythagorean identities now I did tell you guys that you know in the United States of America months high school students very few high school students I've ever seen the diagram or ever see this diaper well they're told in Geometry to memorize this this last year and then what trig teachers in Algebra 2 teachers tell students is okay remember this from last year science fair plus cosine squared equals one you probably were told this was true last year using the old style I think about this old style sohcahtoa / H squared plus opposite of cosine is a over H square and then your geometry teacher may have told you this is true because that means o squared plus a squared over squared equals 1 and why is this true hopefully some of you saw this last year new John's reclass it's like why is this true now see what I did is I use Soca and then I guess I'll then and then I get to hear and I'm like yeah why is that true no one knows yeah what what part of Saturday and be more specific I can replace o squared plus a squared with what because the Pythagorean theorem this is true because o squared plus a squared is hypotenuse Square correct this is hypotenuse squared right opposite leg squared plus adjacent leg squared makes hypotenuse squared so you got H squared / 8 squared which is 1 so you were probably told this last year and now this year I'm supposed to tell you you can create these other two by being smart we're not going to like most students never see this right so we come up with these a different way what can you do to this equation right here to turn it into this equation did anyone see what you can do what can you do to this equation to turn into this one what Miracle I'm going to perform a miracle right now we're going to turn this equation into this using algebra what are you doing turn this now it's not it anyone how do you turn this equation into this equation using algebra okay and then I'll give you a hint hinchas you're going to divide this equation both sides of the equation by the same thing you know how you can do that right cuz it's division property of equality as long as you divide both sides by that same thing you're okay so what should I divide both sides by name wrong answer three times it's divided by what what will turn this into this you guys are on the right track but you're not going to sign Square that's correct divided by sine squared divided by sine squared 1 divided by sine squared cosine squared over sine squared cosine squared over sine squared would be then finally divide this by sine squared what's 1 over sine squared of course it's cosecant squared do it again what would I divide Best Buy to get the last one here you know this is true you could divide this by something and get this Jonathan crap Anthony correct you could take and divided by cosine squared sine squared divided by cosine squared would be chance of squared cosine squared divided by cosine squared is one in one divided by cosine squared is secant squared so throughout this country the way students learn this is a learn this in geometry because of this and then they learn that you can rabies Other Guys by first dividing by sine squared and secondly divided by cosine squared I have to speed up here boy let's look at these ones first when using pi over 2 minus Theta when you text Ada and Dupuy 5/2 - 8 what are you finding when you do high over 2 minus Theta what do you find in when you do pi over 2 minus Theta what are you finding don't don't look at this what do you sign in okay no one's give me the answer so I'm going to give you a hint High Ever choose 90° you didn't 90° - A angle in degrees what are you finding Anthony correct or incorrect you're finding the complementary angle so pi over 2 minus an angle is how you find the car I already told you guys what you see here which is finding the sine of a complementary angle is what cosine is that's what cosine is cosine stands for complimentary sign likewise over here what's tangent of a complementary angle called complementary tangent these guys are called the co function properties these are called the co function properties the fact that you can take sine or tangent of a complementary angle call it Elementary I have one missing here if I had a secant of pi over 2 - 8 pirate 2 - 80 equals cosecant over here these are what book you're looking at I learn these as the Odyssey van identities the odd-even identities I don't think your book calls not even I didn't but I remember them as the odd-even identities sign in Tangent are odd functions while cosine is an even function why is that significant What's significant is about even function make negatives disappear what is the most basic even function what is the most basic even function even function the first time I ever use the phrase even function I said look at that even it look at Weis called even Jonathan crane x squared x squared is most basic even function what does x squared do to a negative it gets rid of an egg mine is just like a square when you do cosine of negative the negative disappears on the other hand what is an odd function do to a negative it does not get rid of it excuse odd function when you cube a negative number you still get a negative number when you do something to a negative number you still get negative when it's odd and Tanner are odd functions negatives cosine is an even function negative disappears so here's an example of how you do these problems there are two types of problems on your homework first time to track where your just told II any problems where you're told to verify the only difference between these two types of problems is when your child to simplify you don't know what the final answer is you don't know what the final result is going to be when you're asked to verify your given the final results so you're you already know what your desk this is how the problems going to end they do miss at the end of the homework because the problems were they ask you to verify tend to be harder we don't want you to give up so they tell you the answer if I just tell you simplify this you'll probably give up on the problem can you start doing the problem you'll be discouraged can you count to give up like you start no problem like ears with the work I'll look like you start doing work start doing work and you like I don't know where this is going and maybe even after this step you'll be like I don't know where this is going. So verification problem tell you what your final destination is so you don't get discouraged this was an easy problem tangent of negative beta just went over the tangent Sade so I can take the negative input in front next tangent sine over cosine so I ride it a sign over cosine that means pocosins can cancel and I'm done negative sign very easy problem finish. Very well on my screen here that's how it looks so all I did is I used an odd even identity to make the negative go in front when I use tangent sine sign Google cancel playing in Beaumont done trust me it's easier to do calculus on this Benihana's I can find the derivative of this a fraction of a can do that here find the derivative of this would take me a couple minutes that's why we learn this stuff so that we can make that goes easy for a cell example cotangent is cosine over sine now unfortunately this from a little harder way sure would be cool if I replace this and then suddenly things would cancel so all I can do is combine the two tow signs together and I get this now this point it's very easy to get discouraged cuz you're looking at this I don't know what to do with this now think about what you would do if these were numbers you might try to combine these two that one has a denominator of signed data this one doesn't have it on meter as number one how would I combine fractions and fractions are being added as to make a common denominator so that's my motivation for the next step and look at the next step and I go I need a denominator sign so I'm going to Malta Best Buy sign over sign member sign over sign is one so I can do this * sign over sign probably should show that work so what I'm doing here is as I'm going to multiply buy sign over sign that's what I'm doing here x sign over now I have a common denominator so I can combine him but look what I get when I combine them oh my gosh sine squared plus cosine squared which is one so whenever sign is I can't overemphasize how much easier it is to do calculus on this Phineas on this hard to overstate that so that's why we learn how to do this you're not a problem last problem last example this one's super difficult so they give you the destination so why is that helpful any guesses on why it's helpful knowing what the final answer is I told you one reason one reason is is to make it so you don't get discouraged on harder problems but it also enables you to do something just remember when you were doing proofs in geometry of course we'll be told what you have to prove and then I know I didn't do this I would take the thing I'm trying to prove and I would right at the bottom of my proof like this is my destination Matthew there we go answer my question correctly attracts going to allow me work backwards allow me to work backwards just like you did sometimes on geometry proofs I hope you did I hope your job in CT first thing I did hear is I wrote this okay you don't know this but trust me okay the second thing I did is I came over here and wrote this and then Rose desk so the sequence that I did this problem is is I wrote this got a little bit discouraged so I came over here and I work backwards tangent is sine over cosine men Signing Time sign a sign so I got to hear and when I saw this and then I wasn't discouraged anymore because I noticed one of these have in common when I went backwards a little bit really easy steps Deacon is one over cosine that's like a no-brainer tangent is sine over cosine and now because of the two signs okay I can combine them Acura correct so when I saw and I saw this I knew I was going to be because I know both of them have a Sakura pointed out he's the only one Apparently that is so I is all I have to do is get this guy to have it so I did this by course again I X a fancy version of wind i x sine over cosine and now I got a common denominator cosine and then I was like here and here and I'm like wait a second one minus cosine squared sine squared how do I know that now by the way a lot of people fake these things they're like I know these are supposed to be equal to each other there for I'm just going to go ahead and write geometry proofs really know the reason why yeah Anthony because the Pythagorean identity one minus cosine squared because when you take sine squared theta plus cosine squared theta equals 1 and you move this guy over this side you end up with sine squared equals 1 minus cosine squared theta so that's why these are equal to each other obviously this is pretty complicated but if you taken and small step it's rocket science you know I saw steak and I wrote one over cosine and sine over cosine and then I saw that I can combine these in the sine squared and then I just look at these two I go hey I'm encouraged I see a pro sign in denominator here I should cosign denominator I can get these two have denominator cosine I get to hear and I'm like that's right why is it right because you took this bag and identity you know that one minus cosine squared does end up being sine squared so that's it K ran out of time I guess I'll be back here on the Rite Aid

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