 how in pre-classes and worksheets there's going to be sort of a lot of isolated techniques and sort of formulas to remember and sort of the whole point of the rest of the semester is going to be at least in classes when we're not talking about projects to try to capture these in some unified framework where we don't have to memorize so much, right? If you go and pull up Google and look for like trig identities, you'll see just like a million of them listed out and our goal in mathematics is not to memorize, you know, a bunch of formulas like we want the the processes and so the processes that sort of govern and sort of the processes you could use to derive these things and then sort of a few basic principles you can just derive everything from. So this is like a technique in math that goes back to Greek era like Euclid, you know, he sets up geometry with four or five axioms and then deduces all of these theorems about them and so like the core principles, the axioms are the important thing and then the processes used to get all of the results are important and like somehow just memorizing all of the results isn't like not the end game. Yeah so just with that being said the stuff I'm going to be talking about is going to be slightly different in terms of notation and phrasing and the stuff that you're looking at, you know, in the videos and pre-classes but hopefully it helps tie it together. So yeah today we'll talk a little bit about what's in those and then we'll talk about the project. So the first thing to say a little bit about something we have sort of already looked at, centric functions as geometric ratios not rations ratios. So the idea here is that we, so we've chosen some coordinate system as usual, we have this like y hat direction, we have this x hat direction. The way I like to think of this is this is like a frame of reference if you're doing you know something like physics then you know it kind of depends on where you like what reference frame you're looking at. So I could go out and draw this and chalk on the sidewalk and I'm just arbitrarily chosen here's the place to put the origin but the point is that you know we make this choice sort of arbitrarily but all the geometry we can do won't won't really depend on that choice. So we have a spot that we're just calling the origin no and let's just say we're given some angle theta and we remember that this determines a ray until we right so this thing starts at the origin just shoots off to infinity and let's say we also choose a radius if we've chosen a radius then this normally this would just continue off otherwise but if we choose some radius R then we get a well defined honest point in the plane and so this is kind of the idea behind polar coordinates that you can specify you know something in the plane like the plane is two-dimensional right you have sort of two directions you've chosen the x hat and y hat so somehow you need two pieces of information to tell you about where you are in the plane how far are you along the x hat direction and how far are you along the y hat direction but it's also equally good if I just tell you so that would be like an x and a y it's also equally good if I tell you just start at the origin turn in some direction and go x or go r many feet in that direction then you can still sort of get to that point and I want to point out the thing to sort of this sort of a picture to keep in mind here and switch over to it this is kind of the thing just to convince you that every point in the plane can sort of be specified in this way so here is sort of you haven't so you're in the x and the y plane but you're specifying an r and a theta so it's kind of like you're standing at the origin you're casting out a little ray in some direction and then you're asking go our units along that ray and I just want to convince you that it's plausible that you could hit every single point in the plane this way because if you shrink r and grow r you could make this sort of anything along that ray you want and then if you vary theta you can sweep out a little circle you know this from zero to two pi so you can hit everything there and if you just make the radius a little bit bigger you can hit everything there so you can kind of it's not really approved for anything like that but you can hopefully convince yourself that you can get every point in the plane this way okay so we get some point call it P with the vector symbol on it because we're thinking of this kind of as an oriented it's something that starts at the origin and goes to that point we're just kind of arbitrarily making that choice and this is getting into the idea that we're now thinking of these not it just points in the plane but vectors something as a you know it's an arrow it's just in the plane it has a magnitude r you know a length and it has a direction theta and maybe well theta or this P really depends on theta and it kind of depends on r as well since we saw in that picture if I vary P and theta I get different actual points that ends up on sorry if I vary theta and r so the point here is that you can sort of do a natural geometric operation here we can drop a perpendicular line to the x-axis it's just forming a right triangle there and we can think we can have you can take this angle and we can think about an adjacent side we can think about an opposite side and I pot noose so formed by this triangle this is hopefully familiar nothing too new here but the the new piece of information is that if I give you a point in the plane or in polar coordinates I don't specify anything else you can always just extract a natural triangle out of it in this way and so you can think of things like sine of theta I want to right here sine of theta in this case will always be opposite over bot noose cosine of theta it's the adjacent over the bot noose tangent of theta I guess this one is so it's sine over cosines the H is cancel and you can over a this is for those that haven't seen it before there's this like so katoa thing to help remember these and then if you do there's all these co-things there's it's always hard to remember which one goes with which so I think it's cosecant goes with sine secant goes with cosine and cotangent just matches up directly and the way you remember these is you just take I guess is an important thing I'm leaving out here so there's kind of a warning the thing to remember here it's going to be pretty important later is that there is this symbol kind of floating around this sine inverse and so I'll tell you what that's actually defined to be it's defined to be this arc sine function of theta so this is a place where notation can be kind of confusing because we use this to the negative one power to indicate taking a reciprocal like exponentiating something for trig functions and specifically for trig functions this notation indicates like an inverse function so you maybe talked a little bit about a lot of different types of inverses you can have inverses that are like multiplicative inverses where you multiply them together and they cancel and that's like a reciprocal you can have functional inverses which we kind of looked at in I guess unit two right where you kind of compose them together and they cancel it's like exponential and log or functional inverses but these are generally two different two different things and so what I will write in general like if I'm doing something like you know just say here sine inverse of theta is not equal to one over the sine of theta and so this is a big caveat and how might you remember this well the only reason we kind of get away with this kind of weird notation is that we already have a name for one over sine of theta which is go see again right it's literally just this thing here and so that's that's at least one way to remember remember that so I'll just move this over a little bit and then similarly this sky is one over cosine of theta and if you want this is one over tangent of theta and so you know to get these these ratios you're just flipping the orders that you see on the left-hand side taking reciprocals to be more precise okay and then for these if you want there's another sort of mnemonic device and I don't know how you would even begin to pronounce this mucosa cow chaos of you like that just something silly to help you remember like which which order you need to take these ratios and then the point is here is that you can actually do this with any point in the plane you can ask about at least geometrically this is something that sort of scales out with the triangle so like we are restricted to the unit circle necessarily if we know all of the sides of this triangle so like if we knew this p bar point in terms of x and y coordinates then we could figure out what these side lengths are and we could just take these ratios and that'll give us a way to compute sine of theta but right if we if we just know theta this is going to be a little bit of a problem so this is kind of a problem that just having theta isn't enough this really is kind of an important idea that this this data is only determining a ray so we can't really make one I'm just kind of going back up to this picture here we can't really make one triangle to try and take these ratios so you have to sort of pick you know like a convention to go with pick one triangle if you want to just define so I guess this is the problem of given you know theta what is say just sine of theta and so the way you solve this problem is that if you're just given the angle you pick a special radius just a radius of one and so you end up with a picture see something like this this is supposed to be kind of an analog of the picture up there you have a y hat you have an x hat this thing is now the unit circle this is a radius of one you just remember that if you pick some theta and you cast a ray out to infinity like that then all I'm saying is that well you didn't choose an R but you can choose a very special R like R equals one and you can ask about this point where it intersects this is like a totally geometric way to define what the sine of an angle is if we don't know things like sign links where you don't have a triangle lying around so we know that this will be some p bar depending on theta we've kind of picked a special R equals one and through some geometric principles this will be an x y coordinates cosine of theta and sine of theta and really I could have sorry keep scrolling back and forth but if I go back up to this picture here I could have also noted that in polar coordinates this is given in the x coordinate it really is our cosine of theta and our sine of theta and so this is really just a special case we've taken R equals one so I'm trying to like fit this all in there if I can so this is essentially how you would define sine of theta cosine of theta cast a ray at that that angle go out and look for where it intersects the unit circle and then just take the x and y coordinates of that thing okay so I want to do a slightly bigger version of this picture so we can talk about the components of this thing well maybe let me do this let's kind of zoom in on like just this part here and look at this situation we have something like this and we were intersecting some unit circle at this point right so this is literally just taking that picture above and we're just kind of zooming in on the first quadrant we still have this angle theta we still have this direction y hat in this direction x hat and this p bar actually let's do it this way let's say that you were say that this thing is the unit circle and this thing is just an arbitrary circle of radius R so right we have this this ray determined by theta and it you know keeps going off to infinity and we know that in polar coordinates this thing is given by it's called this p bar zero and this p bar one this one was just cosine of theta sine of theta and this one is kind of a scaled up version of that our cosine theta our sine of theta and let me shrink these down a little bit so we have some room to work now that didn't go well okay so what I want to ask at this point so if I have polar coordinates for a point maybe off to the side here oh so recall that we have this there is an addition for vectors so anytime I have two of these vectors and the way to kind of remember it is tail to tip these are purely geometric objects in the plane and I'm just defining something I'm calling addition so it's some weird operation it's nothing like addition for real numbers we're now adding two geometric objects but there are good good reasons to you know it shares enough properties of addition that it's useful to call it that and the way it works is that if you have one arrow like this and another arrow like that and this I'm not going to be able to record in the notes I have to just kind of do it in real time here the way you add these things is you just right and so you're in some coordinate system you've chosen these are just two random arrows they could be anywhere sitting around in in space you can kind of pick one up the one you want to add you move it so the tail of this arrow is at the tip of that arrow and the resulting arrow you get starts at the bottom of the first one and ends at the top of the second one so if this was a and this was b and the thing in red is the thing we're just calling a plus b that's just an aside so what we want to do here is kind of reverse this process we're given some arbitrary vector and we want to break it into components and we'll see that this comes up in the project oh well probably I won't say too much about it now but if you're thinking about things coming from physics you know maybe if you're working against gravity or something then you're moving in some some vector some direction three-dimensional space now and somehow you know only the force that's pointing straight down really matters so somehow you have a vector but you want to decompose it into like which part of the vector is pointing down which part of the vector is pointing sideways that kind of thing so that way you can you know determine which ones influence like a calculation you're doing and which ones are kind of orthogonal to it and not contributing on the project we'll see this because you'll have this leaf pointing up you'll have a light ray coming in at some angle a kind of the only contribution of the light ray to the energy going into the leaf is the contribution that's going straight down so you decompose this light ray coming in at some angle into like a proportion that's coming in straight down and a proportion that's coming in sideways so it may not make too much sense now but hopefully it will as as you spend more time with the project so the way what I want to do with this vector here is decompose it into two things I want to decompose it into kind of one piece along the x-hat direction and one piece along the y-hat direction so in other words I want to use this kind of vector addition law that I have to get some vector down here and okay there's not great notation for this so let's call so this p-bar was actually the point out there let's call v-bar this vector starting at the origin and ending at that point so we can kind of go back and forth and I didn't identify one in the other but they're kind of two different things really one of them is just a point in the plane but we know we can always just get a vector from it by connecting the origin to it so we have some vector v we want to get some component v you know just put v sub x so this is just like what is the horizontal component of this if I want to get to that point how far you know east do I have to go if I'm just walking directly east and I want to get a vertical component like this this would be something pointing that way I think of this as like a v-bar sub y so some vertical component that gets me to that and the name of the game is going to be and I want v-bar as vectors where we're not thinking of this like vector addition law to be v-bar in the x-direction plus v-bar to the y-direction so the x component or the y component of this vector and right this is this is just coming from the geometric picture I notice that if I add the red and the blue and do the tail to tip thing I start at the the the origin is kind of the the entry point of the first vector I follow it along I go all the way up to the end of the second vector and I end up at the point I want so this is satisfied by these geometrically and the question is how do you actually get formulas for these things and actually it ends up being not too difficult really this idea of polar coordinates does a lot of work for us it's really the most most useful concept to to know here so there's actually a way we can just write down v-bar of x okay so a vector right it has to be something that starts with the origin here we wanted to end at this point here so really this is just saying it's what point are we going to end at well we aren't going to go any we're not going to travel any distance in the y-direction we're just going to stay completely level and how far out are we going to go well we're just going to go out exactly our cosine of theta units or in other words if you like this b-bar was just equal to some point x y and so this point down here is just x zero and so that's that's all we're really doing is we're just picking off the the x component of this vector b-bar y so fortunately a similar formula works but it's harder to kind of see why so the idea is well now we're starting at this x zero point down here and we want to get to the point that we want eventually so what do I need to add to it well I don't want to travel anymore in the horizontal direction but I need to travel some some units of y vertically and so I just need to go r sine of theta units up and this is a little bit confusing we think because this point you know it's not zero r sine of theta it's our cosine of theta r sine of theta so like how does this v sub y here correspond to this v sub y we had and the kind of magic of vectors sorry my cat's in the way the magic of vectors is that we have this in physics you might call it like a superposition principle where you can kind of move them around freely so a vector is determined by its magnitude and its direction but it doesn't really matter where you kind of place the the base of the vector which means that we can kind of pick this whole thing up by a copy of it and I can just move this around the plane freely and it's still the same same vector so one thing I can freely do is move it to a very nice place like the origin and this is literally literally the same vector so this is a little bit tricky right because it's not it's not actually the same line segment at all right they start and end at two totally different places but in this this framework of vectors we're allowed this freedom to kind of translate these vectors all around and what happens here is that it's actually this point is zero r sine of theta or if you want just zero y whereas this thing was x zero and so you can kind of trace this over and even if you want sorry this diagram is going to get pretty busy in a second but you can also just move this up here get a copy of that vector right so there so I've kind of two paths of getting to this point one of them is more useful to figure out the x coordinate one of them was more useful to figure out the y coordinate but you can see just geometrically if I add both of those vectors it kind of doesn't matter which which one I I do if I go around the bottom part of the square I'm still ending up at my destination I go through the top of the square I'm ending up there too and that's just following the this like vector addition law okay so and I will post these notes up later for whatever you know so you can use them as as a reference if you need but what I want to say here is like the takeaway from this is that we can break vectors up into components this is a technical term and the the kind of heuristic or the slogan you want to keep in mind slogan is cosine kind of looks like the x component you can also think of this as like a direction if you want in which case this would be an x hat sign kind of corresponds to the y component okay and we'll just say one sort of nice thing that falls out of this really nice or sort of really easily just do a quick sketch for this one here we're just going to be on the unit circle here are our directions way ahead x hat so I'm drawing this kind of picture a lot because for a lot of these problems this is you know the picture you want to draw um you just think about casting array some angle theta let's ask ourselves this is r equals one if we go to this point it's a polar coordinates is telling us sort of sorry cosine corresponds to the x component sine corresponds to the y component and r is just one so those are literally cosine and sine and so this the length of this vector here is r and so one really nice thing that comes out of this is that um if you just do drop this perpendicular make this triangle you have a right triangle here where the length of this let me just do it this way so there's an a there's an o and an h so these are just lengths of those sides the length of a from that previous discussion or is just the x component of this vector so it really just is cosine of theta okay and the length of that vertical component is the sine of theta and we know fortunately Pythagoras did a lot of work for us that o squared plus a squared equals h squared and I guess we also know that h is equal to r is equal to one in this case so one nice thing we can conclude here is that while I have another name for o it's cosine of theta squared plus I'm sorry I guess I put these in the wrong order so that's that's a um and then here I have sine of theta and I square that whole quantity it's equal to h squared which was just one and I mean it's a and then notation you've probably seen we put the two on the cosine squared of theta it's just equal to one and so this is like the fundamental trig identity and so this just comes out of so this is like some combination of the Pythagorean theorem and the fact that um if you're on the unit circle and you just take polar coordinates of your points the x component and the y component literally just our cosine theta and sine of theta and then I just wanted to point out something here too just kind of be careful about these two notations maybe I just put a warning over here just to remember that well maybe this is the one to be more concerned about if you have this two over the cosine there it's a little bit confusing because this is like also where we put the negative one when we're indicating um you know like the functional inverse so you can't really these these mean two totally different things so the way I would get around that is just try to try to use try to use this um notation if you can it's a cosine of theta so number one is that theta is always a function so use uh perens to to close the argument up so you know like what are you actually taking cosine of and then two is just put the whole thing in perens and just say say exactly what you're squaring um this kind of gets rid of any ambiguity about what could possibly be happening versus something like cosine of squared of theta or something like that so that's not not great notation because it's not clear like if you had other stuff coming after this like plus three are you taking cosine of theta and then adding three are you taking cosine of theta plus three yeah so you definitely use those and stick with these notations of arc sine versus sine inverse I think this is just an easier way to way to go just so that way there's no ambiguity at all you can sort of just put an arc in front of it and the arc is just indicating it is the inverse function of whatever you're you're asking so you could do your arc tan the arc tangent arc cosecant you know kind of anything you want but the arc is a little bit nicer than the negative one okay thanks yeah we got to say something about the project so let me pull up where see how far we got last time um maybe it is helpful first to look at some pictures so let me just pull those in rather than redrawing these I found some nicer diagrams online to kind of say what's going on a little bit nicer um sorry if you remember the sort of premise of this project is actually maybe maybe it's even better to see I have a different visualization first this thing is probably the one I want yeah okay so the setup for it was that okay we're thinking about the sun orbiting the earth you know maybe the earth maybe this is this sphere here that uh we're on and we're thinking about a little point on the earth and a little leaf and we're trying to ask questions about how much energy sunlight leaves photosynthesize they convert light energy into chemical energy and we want to ask some questions about like how much energy are they absorbing if we somehow know the energy output of the sun and so this is kind of a geometric problem because it depends on a lot of things like namely the angle that sort of the rays are coming in from the sun and hitting the leaf and also kind of what angle is the leaf at it'll depend on some other factors like what is the area of this leaf um what is so I guess we're calling we're going to call it the energy density output of the sun so it's going to be some number of like some wattage is how we'll measure power but it would be wattage per centimeter squared so like some this is a you know a density so that way we can scale it up by the area and so this is kind of the situation we'll have is that we know that there's a periodic nature to the orbit of the sun or even just I guess sorry there's a periodic nature to the orbit of the earth around the sun so I guess this picture is a little bit backwards for this visualization anyways but we know that say if I'm just fixing myself to be you know this point on the sphere just because of the way that this rotation is happening I'm going to go from light to dark and then back to light and it's going to follow some kind of periodic pattern where the period will be 24 hours right as the earth just spins and so you can imagine you know if you're just looking at this little square of light being bounced off of the sphere here this is the brightness of that square is indicating somehow the the amount of energy hitting that spot you know if a lot of it is reflecting off and the reflection is going to be really bright and this is exactly the same thing as you know a little leafing you know absorbing a lot of energy we're just displaying it as a reflection here and so you can see that kind of as the you know if the sun's kind of way off at an angle you're getting kind of less light if the sun's like directly pointing towards it then you're getting a lot of light and if the sun's you know somewhere off behind it then there's no light at all to bounce okay so we're going to try and simplify that situation and analyze a little bit about what's going on so the way we'll we'll model it is that we'll take this light source to be the sun we'll consider let's draw it in orange this point here this will be our leaf and we're going to model our leaf as like a little plane or really if we're looking at it from the side just a little line and this is kind of the geometric setup we have is that this L is indicating that's a little bit weird because the light is coming into it but we can sort of indicate it as a vector in either direction just kind of flipping the direction so it's kind of more convenient in this picture to make it point out back towards the sun and we we're going to have a leaf with some orientation to it and that's kind of determined by this end this end just always sticks out of the plane of the leaf so if you can tilt the leaf a little bit this end tilts it some direction and we don't have to worry about these two this is more for a reflection that kind of thing which we don't have to worry about here but what we're most concerned about is this angle here um you call it theta one in this angle here theta two sorry I think I've labeled these backwards in my notes so let me just call this one one and this one two put a one there two there okay so theta one is measuring it's kind of like forget about the leaf entirely just kind of fix the plane that the leaf is on and ask yourself where is the sun along its orbit like what um what angle is it making with the ground so that's going to be one piece of this puzzle just like what angle is the light source at from your measuring place theta two is kind of the one we're ultimately after because this is going to determine how much energy is absorbed the idea being that if if theta two was equal to zero then this light vector and the way the leaf is pointing would be totally aligned and this would essentially mean that the like it's like 12 noon or something the the sun is directly above your head all of the light rays are kind of hitting the leaf in a maximal way I mean it's absorbing the most energy whereas if you kind of tilt it off to the side the light source is kind of way down off to the left and all of these rays are coming in parallel and like none of them are hitting the leaf the light's absorbing no energy so both of these um will sort of matter for us here and what's what's happening here let's see if I have a good way so I guess I'll just draw this it is we have this little plane that the leaf is on has this kind of normal vector sticking out and we're thinking about this in a very simplified way by kind of rotating this and just looking at it from the side has this is just one simplification like this was the leaf or at least like one tiny little square on the leaf this was our vector and that and before we had some light ray coming in sort of like that really a bunch of them but we'll kind of move them all so they're coming in the same direction and the way we'll simplify here is let me just put it from this side so it's a little bit easier to reconcile with our previous drawings so it's just whatever um let's see what we call this l I'm sorry I realized the coloring is not matching up so well here okay but you can't see that okay green green will be all for us um so the way we'll simplify this is that we'll flip the direction of this like that and we have this as we call this bottom one theta one this top one theta two and actually I think I want to make this slightly uh bigger the sort of key principle that we'll be using is that once we kind of fix the coordinate system now we're kind of in a two-dimensional world so we can fix an x hat and a y hat and we'll pick an origin namely to be this point here this is kind of like we're now setting up a little reference frame this is where our leaf was sitting at the origin and we had some vector coming out like this this is our light vector and we had uh sorry we do it in a different color so now that the the end vector so kind of the orientation of our leaf is pointing straight up so it's kind of lined up with the y direction for us sorry I just want to make this a little bit smaller okay and here's our theta one here's our theta two okay the the first principle that's going to be important here is that there's going to be an energy density um well maybe I'll just say what it is first and then I'll explain what the picture why it should vary um so this energy density is actually going to depend on this light vector so it's like a function it takes in the light vector and I'm not going to say what the actual formula is I'm just telling you some proportionality relationship between them and what is it proportional to I'm going to use some kind of weird notation but I'll explain what it means okay so l sub y is when we do this decomposition into two pieces we get one vector like this which is an l sub x and we get one vector like this which is an l sub y okay so we've broken it down into what is the horizontal component the l sub x and what is the vertical component l sub y and l sub y is the component that's kind of lining up with the leaf because we've chosen the coordinate system that way and with this kind of funny two bracket two lines on the end this is saying the magnitude of that vector or the the norm of that vector or if you want to think of it as just the length that's totally fine but it's saying that as this so this this green vector l right this is the light vector sitting at the origin now and now you're imagining this vector over time this is essentially pointing to where the sun's coming from so it's going to start at zero and it's going to just kind of run through a rotation right it's just like zero to pi and then it's going to you know be on the other side of the planet or whatever and you're not going to see it and then eventually at the next day 24 hours later it's going to start that again it's going to be at zero and then sorry start rotating through this again so the thing to keep in mind is that if you're completely horizontal then this l sub y that you're seeing is not going to appear or rather it's going to be like a vector with like zero if you're kind of way down here early in the morning then this vector is really short all right and if you're up here in the middle then the vector has some like moderate length if you're at like a 12 noon or something pointing straight up then this vector is probably as long as it can be again we're just looking at the l sub y component of it so in other words like the height of this this point over time kind of imagine that this point travels along a little circle and it kind of gets to a maximum height at noon and then kind of keeps traveling you know this as the day varies and so the principle here is that the this energy density is going to vary and I should tell you kind of what the energy density means this is kind of hard to deduce from the handout so I have a picture for that hopefully okay this is something I pulled from a tutorial online um so yeah the star here is supposed to be the sun but the blue thing is supposed to be our leaf the red thing is the the orientation of the leaf the normal vector pointing out of it um and what's happening here is like in the top picture you can see that as these things are lined up you have a light it's kind of like it's coming out in a ray and you want to ask yourself like this is just some fixed ray um with some like fixed amount of energy and so what's happening is that ray is like hitting a smaller spot with the same amount of energy in the first picture versus the second picture you have the same amount of energy but it's kind of spread out over a bigger area because you're at this angle and you can see kind of how this ray is intersecting your your little square um so this is like small area versus bigger area and these two are just like some fixed amount of energy and so all I'm really saying with this proportionality relationship above is that whatever this energy density is it's going to have to depend on the component of this vector that's that's flying perpendicular to the sun or in other words like where are you on this picture like is it is this vector completely lined up with the direction of the sun so you can imagine that there's um let's see I guess we've done it in green here there's like a green vector coming directly down out of the middle of this and in the first case these vectors are like directly lined up and so that's saying that like the y component is maximal in the second case they're kind of a skew so like the y component is somewhat less and then like I said if you go off to um an angle of I guess pi or something like that it's it's not hitting it at all okay so the thing to remember here is like only the y component contributes and so the first goal is going to be find a function maybe find functions for the norm of l sub y and maybe what it'll be so we looked at this picture we had theta one and theta two and I think what you maybe want we'll want to do is start with first finding f of theta sub one equals dot dot something g of theta sub two equals something and what do I mean by this I mean that you have to kind of go back to the geometry of this picture um and ask yourself if I'm given this light vector and let's say I'm given this theta sub one how do I compute the length here so I claim that you have enough and just what we've talked about today to actually figure this out now it's just it's a little abstract so you have to kind of spend some time um and thinking about vectors maybe reading up a little bit about them on Wikipedia and kind of getting used to working with them um but maybe I'll just show you what it is for theta one and I'll leave it to you guys to figure out how to get a similar formula for theta two right because eventually what we this is kind of the the goal that we really want is we really want a formula for this energy density um but we can start off by just making it a form of formula involving the angle so I'll show it for theta one so we'll need to carry around this picture this is really important and this kind of analysis like we always always need a reference picture for this kind of thing so what I'll do is I'll I'll simplify this picture a little bit because I'm just thinking about theta sub one make something large and in this case I have something like that I have a y hat direction and an x hat direction and let's just say like we're not even in the case of above let's just say we're well might as well just do it so you have this this vector l bar and we want to find what is this decomposition um what is the yl bar sub x well what we can do is you can write the simpler coordinates just like above r cosine sorry I forgot to mark this was theta one r cosine of theta one r sine of theta one again the key key thing to remember here is that cosines kind of correspond to x's y's correspond to uh sines and what we'll do is you can actually there's an important assumption here as we can assume r equals one the reason we can do this is because later on so right now we just kind of want a direction that it points and later on we can multiply in um to kind of scale it up to whatever we need so as long as we have a direction then um we're kind of in business so there's some vector like this and there's nothing too crazy here happening I promise but this is definitely a definite derivation you'll want to sort of explain a little bit in the project this is exactly what we did before we decomposed it into an x and a y direction this is some v bar I'm sorry I guess we're calling it l um l sub x and then in blue l sub y and I know that if l bar is equal to r cosine of theta one r sine theta one well guess what I can just copy this since we have one minute left here try to speed through the rest of this and if we want to take the x component well delete the y component we only want to go in the x direction if we want to take the y component well delete the x one we only want to go in the y direction and so this is the one that we want here and now you just need to know one other thing that if we have v bar is equal to a point x y then this norm of v bar well it's really just exactly the Pythagorean theorem for this thing that's all the norm really is just kind of fancy notation for it and what it is let me just draw a picture here so if you have some vector v bar then this length is norm of v and so the way you compute this length is you take x squared plus y squared and you take the square root just because if this was x y then this would be a length of x and this would be a length of y so nothing too nothing too fancy there and so if you want to take the the norm of this how y well I just need to take the square root of something what is the something I need to take the square root of 0 squared plus I guess I can go back we we made this nice assumption that r was equal to 1 so we can delete all of these r's and that sine of theta squared and I'm lucky for us it simplifies to something really nice it's just the sine of theta and so the conclusion is that this energy density which depended on in this case the light factor is proportional to I'm sorry this is sine of theta one everywhere sine of theta one where again theta one was this angle that the light vector made with the ground so there's still some analysis you have to do with how does that depend on this other angle theta two and I claim that you can just use some centrifugometry stuff from class to relate these but that's what you want to start looking at trying to derive these formulas and I think that's pretty much it for today all right