 Hi everyone, this is Gicho. Welcome to another live stream. Today is October 5th, 2019 and we're doing a drop in math tutoring session for the 2019-2020 school year. This is number four that we're doing this year. Open discussion, let's do some math, math, high school math anyway. Maybe a little bit of elementary, maybe a little bit of post-secondary. Okay. We're going to keep on doing these. Provide a space for people to ask questions and if they need math help, we'll try to do them now. And there's a few people that come here that are well versed in the language of mathematics and a couple that are much better at math than I am. So they've been helping people out as well. Okay. Aside from that, if you have any questions or anything you want to discuss regarding high school mathematics, just post your questions and name of the game we'll deal with math first. But it is an open discussion so we can talk about education, basically anything school related. Okay. Aside from that, it's just a sort of a little waiting game until people start rolling in. And let me show you what I got in terms of snacks here. I got tea. This is like 99% of the time I got tea with me. Re-subscribe. Welcome, welcome. Welcome to a math tutoring session. Drop-in session. Apple and tahini. Okay. Tahini with maple syrup, this one. Mask of Raven. How are you doing? You love these math stuff, eh? Fun. I like it. I like it. Just an hour ago I was doing physics, university physics with a student that graduated. And he had some problems that we dealt, that we did. I watched all your YouTube is never caught a live stream yet. Vicious on YouTube is my, is me. Oh, okay, cool, cool, cool. Re-subscribe. I don't remember your name from here. Especially I don't remember the logo. I guess that's a snowflake, a whole bunch of arrows going in. It looks like a snowflake thing. Welcome to a live stream. I hope you enjoy this version of things. It is basically the same thing as the videos I've been loading on, but it's a little, watching a live is, it's like watching a game live, right? It changes a little. And by the way, the sound should be better now. Okay. For some reason, a couple of months ago, because I was doing a certain setup, I set this up backwards. So I think that's why we're getting a low sound. A lot of people were saying the sound was pretty low on some of the viz that we loaded up. So I checked my settings and did this. I went, oh my God, I got this the other way. I thought it was okay in both directions, but it wasn't. I had set it on the one direction coming this way. So hopefully the sound is better for people. And the lighting is going to change right now. We're in fall. We had like rainstorm this morning and then the sun came out and then we're getting cloudy over and it's getting hazy and then sun's coming out. So it's going to go through ups and downs, right? Intrepid. How are you doing? It's been a long time. I'll see. Hope all is well, my friend. Doing well, brother. Doing well. Enjoying the school year. For some reason, the school year, the summer was really short. It seemed really short for my end and for my students as well. And we started kicking off into doing mathematics with my students right when September started. And usually it doesn't pick up for me until October, but I'm a month early so I'm well into the school year now. And loving the time is spending with my students. There's like three or four that I've had for a long time that I'm working with. And it's a pleasure working with kids for an extended period of time, right? You learn their character, you learn about some of their struggles, what they've come through, and how you can teach them better, right? Which is really important in our education system, which is one of the main problems in our education system where, you know, kids go from one class to another class. It's totally different teachers. And then one year to another year with the same course, totally different teachers. It's good if you had a crappy teacher the year before, but it's also bad because the new teacher doesn't know what you know. They have to go through. It's very weird. It's very weird. Once, for me anyway, once I work with a student for an extended period of time, you can fill in the gaps what they're missing, speak on Zala style, like really depending on where they are in their education, within a couple of weeks to a couple of months, two or three months, you can fill in everything you need to fill in and then you can just build on top of that. And kids, man, people are weird. We're so much more intelligent than what the centralized system gives kids credit for. Kids right now can learn it much faster than we learned. I learned anyway. So it's a different game. Jamie James 101. Hi, Chico. I'm new to your channel. Hope you're doing well. Doing well. Thank you very much. And welcome to our channel. Welcome to sort of chill, multi-layer stuff where you change things up a fair bit. I hope the sound is coming okay now. I see it a little thing on my trigger and the volume is higher. It's usually where it is, but I think it's crisper now with the setup, with the thing pointing in the right direction. Should we do some problems? I wrote that. I worked with a student that we did physics about an hour, a couple of hours ago, and I wrote down a couple of the problems which were which are just basic physics questions and the kids in university. I don't know if I should call them a kid anymore. I call all my students kids. Actually, do I call them kids? I don't call them kids. I call them by their names, right? But when I refer to them, I call them my students or my kids. What color should we use? Is this one? Does this one sponge off? So if you have any math questions, post them. For sure we'll do. I'm assuming things will get more math questions coming in when we get closer to exam period. Let's check this out. Does that come out? That comes out good. Let's make sure it rubs off easy. That rubs off easy. Okay. Should we do a little physics? Should I show you? I wrote it down. It's just a comic book. I'm bored. Extra bored. Here. I'll show you this one. Check this one out. So the name of the game for physics is you got to check your units, right? Units is everything. That's what physics is, is applied mathematics. So if you know math, then you can do physics. All you have to know is understand the system you're in, right? So one of the questions my student had was this. And this is, you encounter this in grade 11 physics, you encounter in grade 12 physics, and you encounter it in first year university physics, okay? Or college physics or whatever. So this is this is stuff that they cover all three years you get them, okay? If you're lucky you get them in grade 11. You get in grade 11. But basically he had this, right? Here's a graph. This is time in seconds, usually. And this is velocity in meters per second, okay? And he had a graph like this. And this is time. I forget what the numbers were, but two, four, six, eight. Let's go 10 seconds. And let's make this, I think these were at five, 10, negative five, right? And the question he had was this. Actually, you know what? He sent me the images. Let me read off the question exactly. That way I don't make a little mistake. Oh, yeah, here we go. Oh yeah, this is the graph. Nice. Actually, this went to 12, I guess. Here, let me erase those. And then I'll close this guy and then open it up again because my computer works overtime so the fan noise will go up too high, right? Negative 10. So here's the question. Figure one shows the velocity graph for particle having initial position x0, okay? x0. And this is x0. Okay, this is the x. Well, this is velocity 0. But it's starting off a position 0 at t equals 0 seconds, right? And the question was this. At what time or times is the particle found at x equals 35 meters? Okay, so let me close this guy off. That way the computer doesn't go crazy. So the question is at what time is x? This thing starts off at x is equal to 0. At time is equal to 0. At what time do we find the position of this particle at 35 meters? Okay, so just imagine there's a position x here and once time you start the clock and the velocity of this guy is increasing per second, right? So there's acceleration here, right? If you see this velocity and this is time in seconds, okay? What you see is an accelerating particle, right? So velocity is increasing over time. So velocity at time 0 was 0. At time 1 was this guy here, whatever that is, I'll say 2.5. At time 2 is 2 seconds is 5 meters per second. At 3, I'm assuming it'll be at 7.5 meters per second. At 4 seconds is going at 10 meters per second, okay? And then what it does, it starts slowing down. At 5 seconds is back to 7.5. And it reaches a velocity of 0, right? And then starts traveling in the negative direction. So if you're watching this thing, this particle or this person, right? By the way, if I miss stuff, Lance Left Hook, thank you for the follows and thank you for the subs if you decide to stop this channel, right? So there's this guy. Let's assume he's running. I don't know how fast, you know, 10 meters per second is, only kilometers per hour that is, right? We can convert it. Maybe we do a conversion to figure out if this is legit too soon. So it's a man. What, at 10 meters per second? Well, I used to run the 100 here. Well, we can do this mental math, right? I used to run the 100 meters in track and field. And the best time I had was like 11.8 seconds or something, right? I think the best time in the world is, I don't think they've broken 9 seconds yet, like 9. something seconds. I don't know if it's reached 8. something seconds yet, right? It will at some point, I think, but no one thought they would have broken 11 decades ago and then 10 decades ago. And no one thinks that, or when I was running anywhere last 15 years, they will break 9, but pretty sure they'll break 9, right? So if you're going to run 100 meters, 0 to 100 meters, let's assume in 10 seconds, then if someone's traveling at 10 meters per second, that's a legitimate speed someone could be running at, right? Okay, so we could make it a person, okay? So this person starts running from x is equal to zero. This is their position, zero, right? Equals zero, starts running and they're running faster and faster, right? When they get to four seconds, it takes them four seconds to reach a velocity of 10 meters per second, and then their velocity decreases. I should make this a little bit shorter, right? So they're accelerating from here for four seconds, right? Four seconds, and then they start decelerating over the same four second period to zero. So it takes them four seconds to reach maximum speed, right? Richard Zack, thank you for the Twitch Prime, so, right? And then for another four seconds, they decelerate, and usually if you're running 100 meters, when you cross the finish line, you're slowing down, but you can't stop right away. If you stop right away, you're going to break bones, right? You're going to tear muscles. You've got to slow down, right? So you accelerate, accelerate to your top speed and you slow down. So for another four seconds, this person decelerates until they reach a velocity of zero, right? Here's a velocity of zero, and then they start going in the other direction for two seconds, because the velocity is now negative. So if this is our positive direction, and in physics, what you do, whenever you're laying down a problem, you give a certain direction, you make a certain direction positive and the other direction negative, right? It's very directional, it's vector-based, right? So physics, there's a lot of vectors, and the vector is basically magnitude and direction, okay? So this person starts going in a negative velocity direction, and we're talking just a two-dimensional system. So for another two seconds, this person runs back, and this is two seconds, right? Two seconds. So the question was this, okay? And this is, you know, you don't have to draw this, but it's a good idea to know what's happening in this system, right? And that's the key with physics. It's math, applied mathematics, so what you're doing is you're taking your math abilities, the powers that you have, you're looking at a physical situation in the world. This happens to be kinematics, right? Kinematics is bodies in motion, I guess, things in motion, right? You could have a whole bunch of other different types of systems you go into. That physics deals with electromagnetic, magnetic gravity, particle physics, quantum mechanics, anything you want, right? This is kinematics. It's one of the simpler physics. Well, these types of problems, anyway, situations that we have. Sending rockets to the moon is kinematics. It's also got gravity in there and different formulas that you have to end up using, right? So the way it works is this. The question was this, at what point, or, oh sorry, at what times is this person 35 meters away from when they started, right? The wording of the problem wasn't, and that's one of the issues with teaching physics and stuff like this. The wordings of problems aren't the best way, right? If you take the integral of the function of the ground, you get the distance moved in one direction, right? You will, yeah. If you take the integral, you get the acceleration, you take the derivative, you get the distance, right? The derivative kicks you down one dimension or one unit. The integral will kick you up. You get the acceleration through the integral, I believe, right? So for example, if we have here, let me bring up some kinematics. Bring up my kinematics formula, kinematics formulas. And that's what you need for physics, right? You need your formulas, okay? So take a look at this. Velocity and distance, here. Let's do this here. I'm going to do this with a different color. Since you brought it up, we'll deal with it right now, okay? We're going to take a lot of tangents. No, integral gives distance. Oh, integral gives distance? Hold on, let me look at the formula. Integral gives distance. Oh yeah, I'm going the other way. Pooper, me, right? The area under velocity. Yeah, the area under velocity. That's the way we're going to use it, right? The area under velocity. The function is initially a function of speed, so yeah. So let me write down the formulas here. I always get things backwards. So here's velocity. The formulas you have. Velocity final is velocity initial plus at. And the distance is equal to, I'm just going to write it down. V initial t plus 1 half at squared, right? So these are two equations you have for kinematics. And here's the other ones here. Let me write down the other ones as well. Let me bring up chat so I'm not missing anything. Here's the other, you basically have four formulas you deal with in kinematics, okay? Here's the other one. V squared. V final squared is equal to V initial squared plus minus plus 2 a d. And distance is equal to V t. Oh no, we already got that one. Why is it giving me two of those ones? Oh, minus. Yeah, we use this one. Distance is equal to 1 half. 1 half V final plus V initial plus V initial plus V initial times t. Okay. Now back in the day they used to make us memorize these. But I don't memorize these or later on you didn't really memorize these. Okay. Did I forget a distance? Did I forget a distance? Distance right here? This one. Changes in distance. V i t is the constant term, I believe. This one. So basically if you take the derivative of this guy, right? The derivative of this guy is, it's just a quadratic function, right? Thanks for the correction by the way, gang. For sure correct me when I'm wrong. Please, please. It's how I improve, right? So if you take the derivative of this, if you take the derivative of any type of quadratic, here, 5 x to the power of 3 plus 2 x squared, all you do, you kick the power down to the bottom and subtract the one from it, right? Take it down one notch. So this one would be 15 x squared plus 4 x, right? So this one, the power is one. So that kicks down and that becomes here. Let me write it down. Write this guy down here. So D is equal to V initial. One times V initial is just V initial. We're equivalent of it. We're equivalent of it, right? Oh, you want the C out here. That's right. I haven't done this forever. You want the C out here. But we're not going to do an integration. I'm not there yet. Thank you for that. Oh, hold on. How come this didn't get approved? Allow, allow, allow. Sorry about that Putin roaster. The automaton zaps things out, right? So the one comes out, multiplies that, and this becomes T to the power of 0 plus one half times 2 comes down, A, T to the power of 1, and then that just kicks the derivative of a constant, which is what the C is, it's just 0, right? So plus 0 if you want. So this becomes D, the derivative of D, which is velocity. So D, oops, DD over DT, I guess. That's the symbol zone, which is velocity, math explicit. So I can see why it blocked it. Oh, that's why it does it. So this becomes the derivative of D relative to T is meters per second, basically is velocity is equal to V initial and velocity is now V final plus a half kills the 2, A, T, right? That is this. So if you take the derivative of that, you get that. DD over DT. Cool. I gotta erase this now because we want to find the area, right? If you want the, if you want to take the integral, you're going to go the other way and stop, but we're not going to do the integral. Really, I forgot how to do integrals. Okay, I would have to look it all up. So let me erase all this. The brown anyway. Let me kill this guy, clean up our space a little bit. We might leave the formulas up there. So here's what we've got, right? And one thing you have to appreciate with physics is units is everything. It's the units that matter. EHM. EHM. I don't know what EHM is. So units is everything. Okay. So right now, the units of the y-axis is meters per second. The units of the x-axis is time. Never mind. What is this? This is physics, right? He was cleaning his throat. Oh, okay. So yeah, in the distance formula, you would add a constant equivalent to where you start. He's assuming it's zero where I suppose. Yeah, we're starting at zero, right? So x is equal to zero. So the y-axis is meters per second. The x-axis is seconds. So in general, if you get a question like this, where they say at what time is this runner 35 meters away from where they started? Assuming constant acceleration. Huge. Constant acceleration is one of the key factors, right? Now, if you're looking for time, right? If you're, oh, sorry, we're 35 meters, blah, blah, blah, yeah, we do that. If you're looking for a distance, then you look at your units, you go, okay, how do I use the distant property of our graph to get meters out? Well, if you multiply these two guys, the seconds will kill the seconds and you get meters, right? That's one way you can think about the physics problems that you get. Whenever you get a physics problem, look at what units specifically you're looking for or your marker units and take a look at your setup to try to figure out what you could do with the units to get your appropriate units that they are referring to or wanting, right? So for this problem, what you need to do if you multiply those two guys to get the meters you read? I haven't read it yet. I've read parts, but I haven't read it yet and I haven't watched the movie either. I've read the C.S. Lewis, I've read the other trilogy that he had, the more adult sci-fi trilogy C.S. Lewis had, and that I loved. That was amazing, right? They hid the, oh, I forget what they were called. They were trippy. It was very cool. It was very cool. Okay, so take a look at this. So for this problem, the only thing you needed to do is figure out at what area under the curve is equal to 35 meters. I'm not going to lie. Don't lie. The golden compass, the golden compass. The golden compass is Narnia, isn't it? The hidden, the hidden secret or something, and Pallelangelo or something, C.S. Lewis. Too dumb for this. Don't get a single thing about all this. Now check this out here. Daska, check this out. To find out how far this runner has traveled, all you need to do is find the area. Okay, so here, let's find out after four seconds. This is two, three, after four seconds, how far this runner is from where they started. Okay, so if you want to find the area here, check this out. That's an area of a triangle. This is the right angles here, area of a triangle is equal to one half base times height. This is something you do in grade eight. You do this. So this question is a university, first-year university physics question. The only mathematics you need is basically grade eight or grade nine math as long as you understand what the system is telling you, as long as you understand the units. There's a little bit more to it, but the only math that you're really using, you're not even using these guys. We're not. We're not even going to use these guys. We're just going to use area under the graph. I teach high school math, but I'm not a teacher in an institution. I do private. I wouldn't function in an institution well. So let's find the area here. The area there is going to be four times 10. So one half four times 10. One half of four is two. Two times 10 is 20. 20 meters. So in the first four seconds, this runner has traveled 20 meters. Let's figure out how far this runner has traveled up to eight seconds. So another four seconds. This is again four, the total distance from there to there. And again, the height is 10. And that's a triangle again. Well, it's the same thing as this. So from there to there, decelerating, that's another 20 meters. So here's 20. And if you count it to here, that's 40 meters total that the runners traveled in eight seconds. Now they were asking, the question was, at what time or times is the runner 35 meters away from their original position? Well, their original position was here. 35 meters would be like, if that's 40, it would be like here. This is 35 meters. Oops, 35 meters. So the runner is 35 meters away from where they started twice, both on the way here. They hit it once. And when they're running back, they hit it twice. So there's twice that they're 35 meters away. So all we have to do is figure out the area of the graph where the sum of the area is equal to 35. So let's check this out. Let's do the area for this six. After six seconds, this part is 20. All right. Let's figure out what the area is in this part. The area in this part is two times five, right? Because from there to there is five. So two times five divided by two is an area of triangle. Area triangle is equal to one half base times height, which is equal to, let me kill these guys. And this looks scary, but all we're doing is just geometry, right? So one half, the base is two times two times five, right? This kills this. So that's another five meters here, right? So 20 plus five is 25. We wanted to figure out 35, but we haven't figured out the area here yet. So what's the area here? The area here is two times five, right? From there to there is five, it's just a box. So two times five is 10. So that's 10. So 20 plus 10 is 30, 35. Oh, six seconds. This runner is at 35 meters away. So it takes him six seconds to go from here, six seconds, right? The kicker is, he goes to the top and comes back. So six seconds is the first time he hits 35 meters away from where he was. And then what we're going to do is figure out when else is he six seconds away or sorry, when else is he 35 meters away? So let's figure out the area here. This was five times two because six to eight is two divided by two, which is five again. So this part is five, right? So that's 40 meters. And then he's got to come back, right? This is negative velocity, which means in the negative direction. Is it true that the math can answer almost everything? Almost. Almost. You can quantify almost anything. How much do you love ice cream on a scale of one to 10? You just quantify your love of ice cream, right? So let's figure out this is backwards from where you went. Let's figure out the area here. This is again five and from eight to 10 is two seconds. Well, that's the same as that, right? So the area here again is one half times base times height. This kills this. So that's five, right? But it's negative five because it's negative velocity, right? So if you subtract five from 40 because that's what the total area was all the way here, right? That's 40 meters that way. Five meters back, that's 35. So also at 10 seconds, 10 seconds, he's at 35. So it took him four seconds to go there and come back again, right? That was the first problem we did. First year university, a month in, he had this problem, right? You make it look so easy. I wish it was my math teacher. It's just, I have the background for it, right? And some of these problems aren't easy for me. Like we did one problem where we got it wrong and then we had to figure it out. We knew the answer and we sort of went, okay, what is it about the system that we're not understanding, right? So let me show you. So is that okay? This is, again, we just used our knowledge of physics, what the graph means, and that's really the knowledge aspect of it. There isn't really any mathematics here. You know that this is velocity that's time and the area under the graph based on the units, meters per second times, seconds gives you, eliminates the seconds and gives you meters, right? And then we use just geometry to figure out the area, right? Math is abstract enough to apply to many, many situations, both real and made up physics, moving around four dimensions, etc. It can't answer philosophical questions like moral problems. It can only quantify, quantify, not qualify. Very well said, the Mask of Raven. But it can get you to a level where you can, with the quantifying whatever system you're looking at, you can hopefully make the right decision, right? This is what, if you watch science fiction, Vulcans, right? What is good? What's the Vulcan say? The needs of the many outweigh the needs of the few, right? Which I don't agree with. I think that's crazy, right? That's extreme form of majority rules, right? But in certain situations it applies, right? And in the Vulcan Star Trek, it comes up a lot where they have to, you know, one person has to sacrifice their lives to save the whole planet or something, for sure that applies, right? You can also conclude that it's eight seconds for the entity, because eight seconds for the entity, because 35 to 40 minus five is four seconds. So half of that is two seconds, 35 to say, oh yeah, that's right. That's right. The only things that I'm from Sweden, so the terms and stuff you guys use make me confused, yeah. And that's the language, the natural language, right? You know how to speak Klingon? No, I don't. I know people do. Let me erase this, let's do another problem, right? I guess we're just going to do physics, maybe, unless you have math questions. If you have math questions, lay it on us and we'll try to deal with it, right? So let's do another one. Here, here's another problem we had. Okay, this was the second problem that we had. Can you prove that right? Someone's already proved it. They collected a million bucks. I think it was, there's a documentary on it. I think someone proved it already, right? I watched the documentary with Raymond hypothesis and I can't, it's about prime numbers. I can't remember. I think it's about prime numbers. I can't remember what it, prove, I can't remember what it is. The documentary is fantastic though, fantastic, tear jerker. Really, it's a tear jerker. No, no, no, no, no. So here's another problem. It's kinematics. You throw a rock up. Rubik, I've got a class where we're using the dual space of a vector space a lot, dual space of a vector space. I have a really hard time visualizing what the dual space looks like. Do you have any advice on how to think about it? Dual space. You mean X and Y coordinates? X and Y coordinates? Let me know and we'll deal with that first. Thank you for creating this educational space. It is helpful to me. I have a love of all sciences, but math specifically challenges me in math and hearing explanations is greatly helpful for me. My pleasure, Merle. Merle. Merle human. Merle human. I did it again. I said Merle. Merle human. My pleasure, man. My pleasure. Riemann hypothesis hasn't been proven yet. It hasn't been proven yet. What was the documentary I saw? I thought it was a Riemann hypothesis. Intelligent blueberry. I'm having problems with math in class. Don't suppose if I mention it, you can help me up. I'll try to help you out in Intelligent Blueberry for sure. But you're in university, so Mask of Raven might help you out as well, but we'll give it a shot. So Rubik, are we talking about vectors? Like two dimensions? So let's assume here, let's assume we have a ball. Must be thinking of another one. Must be thinking of another one. It's the one where the guy, I'll try to find it and post it online, on Discord channel. Okay. So here's, yeah, I'm going to use brown. This guy's dying slowly. It's the space of maps from a space to itself. So you're mapping. Oh, I remember the stuff. You're mapping a vector upon a vector. Let me lay out the first level of it. And then, because I haven't done anything like this for a while, long time, if I understand you correctly, it's basically relativistic, I think. I would have to look it up to remember how to do it again, but let me explain vectors to people. And then we can kick it up a notch. Zeno's theory. Zeno's theory. So let's assume we have a ball. Okay. Let's say the ball is traveling in this direction, and it's traveling at, let's use simple numbers, 10 meters, oops, 10 meters per second. It doesn't have to have unis. You could just, what about it? I don't even know what the Zeno's I thought this was. 10 meters per second. Okay. Now, assume that there is, we want to find the final velocity of this thing. If there is another force, let's say the wind is blowing. So this guy here was a gigantic ball, kicking the ball at a goal post. Here's a guy, goalie. And this guy's mental calculation is trying to do it because there's wind blowing in this direction at, let's say, two meters per second. Okay. What's the true direction? Where is this ball going to go? Because the goalie is going to try to figure out where it's going to go. And this guy's hopefully compensated for the wind, so you can get it in the top corner. So you can score a goal. And they can give balls different types of spins. In pool, billiards, we call it English, you get a twist and the ball goes like this. So what you do is to be able to do this problem, you draw your vectors, and what you need is an angle. Let's assume this is at 40 degrees, and let's assume this guy is at 15 degrees. Okay. So you draw your vectors, you go this is 40, and this, well not 40, this guy is 10, right? 10, going off the horizontal at 40 degrees. And over here you have the wind affecting this guy, and the wind is two, and it's going off the horizontal at 15 degrees. Okay. Let me draw these bigger so you see them a little bit. Usually you try to make the vectors relative, like if this is 10, two is going to be smaller, right? You wouldn't make the two really big. So this guy's two, and the angle is 15 degrees, right? So for you, the only way to be able to do this, you have to break these things down to their x, y axis coordinates, right? So what you end up doing is you draw here, let's put our axes here. You draw your x-axis, you draw your y-axis, and what you want to do, I'm going to bring a different color in here now. Let's bring a different color. Let's bring green. How's the green? Is it really easy? Yeah, that should work. Cool. So what we want to do is we want to figure out what the y component of this is, and what the x component of that is, right? And this ends up being just straight up geometry, right? So this is your x for the ball. Let's call it x, b, and y, b, or b, x, and b, y. That's a better way of putting it. b, x, and b, y. So this is b, x, and b, y. The x component of the ball and the y component of the ball. I can really understand the concept of the reflection using vectors. Can you help me with it after you finish? I can really understand the concept of reflection using vectors. I can't really understand. I was like, what? I can't really understand the concept of reflection using vectors. Yeah, we can do it. We finish this. As soon as we finish this, let me know if I'll put a little sign here, reflections, reflections, and we'll do it. Mask of Raven. I was asking if Raven hypothesis could be proven using Zeno's Paradox. I should think, I should link you to from 2018 that may explain better than I can say. If you can link it up on the Discord page, that'd be great. That way I can take a look at it too later. Sir Michael Atia, 89-year-old mathematician claims to have solved the 160-year-old problem. Really? Raven hypothesis. Check this out. So we want the x component of this. Well, the x component of this, if you use Sokotoa, sine theta is equal to opposite over hypotenuse, cos theta is equal to adjacent, g-second 2Ds, adjacent over hypotenuse, and tan theta is equal to opposite over adjacent. So here's our 90-degree triangle. This is 40 degrees opposite over hypotenuse, adjacent over hypotenuse. So if we want to find this out, this guy here, we'll do it here, and then we're going to erase a whole bunch, make us more room. Atina proof wasn't great from what I've heard from other mathematicians. Oh, wow, you know that, Masquerade. That's cool. So this guy becomes cos of 40 degrees is equal to adjacent, which is ball in the x direction divided by the hypotenuse, which is tan. So B of x, if you cross multiply, B of x is equal to tan cos 40, right? Let's erase this. So B of x is equal to tan cos 40 degrees. If you use the same thing for sine, for this, this becomes tan sine of 40 degrees. Okay, never actually looked at it. Did it use zinus? I got to look at what zinus paradoxes, right? So this is the direction that this guy's going the ball. This is the direction for the y component of the ball. And then we've got to do the same thing here, right? If we do the same thing here, we're going to get the wind in the x direction and the wind in the y direction, right? I believe so. My information could be wrong, though. Keep in mind, I am still learning and absorbing lots of data once. Cool. Fun to do. So this guy is the same type of thing that happens over here. So this becomes two cos of 15, two cos of 15, and this becomes two sine of 15. So let me put this guy here. We'll put a little arrow. Woop, put it here. And wind in the y direction is two sine of 15 degrees, right? But here's a kicker. Here's a kicker, right? We said that you have to give positive and negative depending on which direction things are going, right? Tyler, thanks Tyler. Appreciate it. I love it. I love doing this, right? So if you're giving positive and negative in certain directions, what we're going to do, we're going to say this way is positive and that way is positive for the y, right? So this way is positive for the x and that way is positive for the y. And anything going in this direction is going to be negative for the x and going down is going to be negative for the y. Well, this is going in this direction, so it's positive for the x. That's going in that direction, so it's positive for the y. That's going in that direction, so that's still positive in the x. But this guy is going down. The wind y is going down, right? So this is not two sine of 15. It's negative two sine of 15. Negative two sine of 15. Okay. So what we end up doing is this now. If you want to visualize what's happening, we're taking this vector, putting it here, right? And then we're taking this part, putting it right at the end of it, right? Because they're both going the same direction. This part of it is 10 cos 40 degrees. This part is 2 cos 15 degrees, right? We're going to put this guy up here going up, right? That's 10 sine 40. And then this one is coming down. It's going to be negative 2 sine 15. Okay. So what we end up doing now is hopefully this focuses. There we go. What we end up doing now is adding these guys up. We're going to get a number. We're going to add these guys up and we're going to get a number. Let's do that down here. Okay. So take a look at this. I'm just going to punch in the numbers. I got to do it on the computer. So let me punch these in on the computer. You guys can do it as well. Right? So we're going to do, let's do this one cos of 40, 40 cos times 10, right? I'll show you in the bottom. So this part here, let's draw it here. This guy is 7.66. We're going to take it to 2 decimal places, 7.66 meters per second. I should have just said meter. You can do it per second. Right? And then this part, let's figure out what this part is. Cos 15 times 2. 15 cos times 2 is 1.93. 1.93. So this whole thing is these two guys added up, right? Which is going to be plus 7.66, 7.66. Which is 9.59. So let me erase this so it's not confusing. So both of those added up is 9.59. 9.59. Okay. Let's combine this and the wind is giving it more power. Do you know who Mr. Higarti is? No. I don't. I'm not sure if that's directed at me or not. Let's do this one. This one is going to be 40 sin 40 times 10. So let's take 40 sin times 10 is going to be 6.43. So 6.43, 6.43 minus, because that's going down, 15 sin 15 sin times 2 minus 0.52. So this part here is going to be minus 6.43, which is 5.91. So the total in this direction ends up being, which is really going to be here if you're going to look at it with vectors, because those two guys kill each other, is going to be 5.91. 5.91. Okay. I really struggle with masks because I'm dyslexic. I struggle with reading because I have a little bit of dyslexia as it comes out when it comes to trying to pronounce names and reading things backwards. People have noticed when I read chats, sometimes they have to correct me from what I've read, right? It's just a little bit of a struggle, a little bit more effort from my part that I have to put in. And I've taught people who have dyslexia some severe, some not, um, Grandad, speak up, please. I'm assuming you're not, what do you call it? You might be here just to play. But just in case, just in case you're legit, you've got a math problem because you've got dyslexia, put the effort in. I've worked with students that have had that problem and they excel, they can excel, right? So all you got to do now is just do the Pythagorean theorem, right? Actually that goes to here, I guess. It goes there. It's ASMR. And headphones are your friend. Sokotoa, like, yeah. It's supposed to be chill. It's supposed to be chill. So if we do this, try to figure out the magnitude of this thing. We're just going to do Pythagorean theorem. A squared plus B squared is equal to C squared. So it's going to be 9.59 squared plus 5.91 squared is equal to C squared. So let's just do that. Okay. 9.59 squared. 9.59. I failed math because of squared. Why did you fail math? Plus 5.91 squared equals that 126.90. So this becomes 126.90 is equal to C squared. So C is equal to the square root of that, right? Because of what did you say? Because of circles. I got an 8. You got an 8? Let's check it out. Square root that. I think it should be more neat. You got 11.26. 11.26. 11.26. That's the magnitude here, right? So the ball is traveling faster than this guy kicked it thanks to the wind power, right? Now you could figure out what the angle is as well, right? Because the angle is going to be going down now. It's going to be less than 40 degrees. And the way you figure that out is you're going to use Sokotoa again. Okay. So the way you do that is let's just use 10, I guess. 10 theta is equal to opposite over adjacent, which is 6.43 divided by 9.59. So theta is equal to 10 inverse. That do it. Keep whatever that is. Let's do it. Could you help me with topics 12.11 of the A-level spectrum modeling with the ventures? Because I'm really struggling with them. Harvey, I couldn't help you on that. I haven't done that stuff for a long time. Modeling with differential equations or integration and stuff like this. At some point, I will get back into it. We will definitely hold live streams for it. But right now, specifically, just focusing on, I know, brother, I know. You know how many people have asked me to teach them calculus? And I will at some point, but I can't do it right now. It's too much on my plate for me to go learn, relearn calculus. 33.84 degrees. 33.84 degrees. So this angle now is 33.84 degrees. So the guy kicked it this way. The wind is pushing it this way. So the ball is actually going to go according to my diagram. This angle, the true motion of the angle is 33.84. 33.84 degrees. And it's going to be traveling at 11.26 meters per second. Can we go back and watch your whole streams? Yeah. I had, I mean, I'm just going to leave this up so you take a look. I'm just going to catch up with the chat. Dang, yeah. Am I on the good or naughty list? I have no idea. I don't have a good and naughty list. You're either in my class or you're out. I had that issue in school too. So you would just like to do completing a square? We have completed a square. A lot of, I've done a few videos on completing the square, by the way. If you do Chicho completing the square, we can do one. But I want to do reflections right now because I forget who it was that asks for it. The GCSE exam board always increased the difficulty exams. In my mocks, I was getting Bs constantly. I only got a C and it was extremely tough. Can we go back? Yep. Yo man, I respect your love for math. Good stuff, Mike. Hey man, I hate English. What about you? Writing in the same paragraph over and over again is a nice thing. Mike, I feel you. I did poorly in English in school. I really did. And at university. I wasn't good at English. But when I grew up, when I got older, I realized that I had to be, I had to relearn English, how to communicate properly. So I started writing. I started blogging. And man, you have to have a pretty thick skin to be able to take all that criticism of posting stuff up and people say, oh, this is garbage. Your writing is horrendous. So I had to relearn English, teach myself. And I reached it to a level where I'm not bad in expressing my opinions. I highly recommend even if you're not enjoying it in school and even if you're out of school and you're not good at it, learn how to do. Learn how to read and write properly. It's really important. Spider beings, how you doing? How's life? Doing good, brother. Doing good. Have you considered putting all of your teaching segments into a sequence and uploading them to a hosting source like YouTube? I'm already on there. Twitch from my understanding removes videos after a time. It would seem to me that you are teaching universal mechanics that rarely change. I could see them used often for further personal reason. Merely human. Do chicho? Go to my YouTube channel. I have, I don't know, 300 plus math videos online that I started creating in 2007. I've categorized some of it. There's, I got playlists where we do mathematics of food and farming. We do mathematics of graphic design. We do a whole bunch of mathematics. We go through the real number set. We start, I start off with the real number set, but I'm going to kick it back a notch. I've got another set of stuff coming up. We've got a whole series on trigonometry that we're halfway through trigonometry grade 12. I got 300 plus videos. We're out to, we're close to 800 videos on YouTube. It's got to be at least 400 videos on mathematics. Check out my YouTube channel and I specifically created a channel called math and real life, a website and a channel called math and real life that you can see. I haven't updated that one. That doesn't contain all my recent or let's say recent last five years of videos of mathematics. And I created another channel called 420 math and a website called 420math.com. If you go there, it has 200 plus videos and that's in supportive organizations that we're trying to improve. Yeah, so I have a lot of content online. I plan on creating modules and teaching all of high school mathematics. We're about a third of the way through, a quarter of the way through. We've got another 10 years, 15, 20 years to go. I don't believe in the concept of thick skin. I think that it promotes a world of unfeeling and human behavior. Me and human, not for me. For me, if I had to grow thick skin because I would write stuff and people would criticize what I wrote, some people unjustly because they didn't like what I wrote. Some people would send me feedback because the way I expressed myself wasn't correct. So initially, I had to be able to filter out the garbage criticism from the good criticism. And that took a little bit of getting used to. I call that thick skin. For me, it is. Because initially I was, oh man, I can't believe I got these wrong and whatnot. I struggle with adding subtracting algebraic fractions. I think it's really cool you're doing math tutorials on Twitch. Yeah, I like it. It's where the students are so I come to them. All right. Lance left hook. Oh, it's ASMR. Do you mind to quickly go through basic trigonometry sometime? I'm ashamed to admit that I totally forgot about it and wiki at the time didn't help when I was in a hurry. I forgot all about cosine, sine, tan, blah, blah, blah. I know it's easy. Lance, it's not necessarily easy. Once you know it, it's easy. But when you're trying to learn something, it's not easy. And it's not supposed to be. Oh, Tyler, thank you. I forgot about my YouTube, the YouTube command. Yes. Thank you for that, by the way. So trig, let me put this on. So we know we're going to do trig as well. Trig, oops, trig. Okay, I'm going to erase this. I'm going to deal with reflections. Are you always this chill? You're kind of the Bob Ross of math. Yeah, I've been calling the Bob Ross of math a few times over the years. Really. Bob Ross of math and Bob Ross of card books and huge respect to Bob Ross. So I take it as a huge compliment. I take it as a huge compliment. Am I always this chill? I don't know. I don't know. I have videos on YouTube. Initially, the videos I loaded on YouTube were sort of graffiti style. I was taking my tripod, big chalk and going out into the city and finding walls and doing graffiti style and mathematics, fast-paced, because I had to be in and out because people don't like gigantic math pieces on their walls. Even though it was chalk, they were like, they were freaking out. I had to take them down a couple of times. I got caught. I got busted. Let's paint the pretty little coast over here. For sure we do. Okay, I'm going to take this down. So reflection. I'm not 100% sure what you mean by reflection, but let's talk about this. Because this is something we use in geophysics, right? When you do seismics and radar surveys and whatnot. So I forget who it was that asked to do this. Reflections, reflections. Okay, I don't know where it is, but if you're still here, I'm going to lay something down. Let me know if this is what you want us to talk about. To me, your speaking method instills a sense of confidence and your knowledge to me. This makes the learning here feel secure and reasonable. I very much like this method of speaking. Thanks, merely human. Thanks, merely human. I try. Sometimes I'm wrong. Sometimes I'm wrong. Sometimes my students, you know, I write down the wrong thing. My students correct me or you guys correct me. I go, oops, you know, erase it. But one thing with learning things is that you have to have the belief. And the understanding is not even belief. It's the fact that if you put the time in, you will learn it, right? So all that's required is to put the time in. Like the bounce of a ball. Like the bounce of a ball. We can do a ball bouncing question. I like those. But let's let me lay down the reflection. And the person that brought it up, I'll put it down here. You let us know if this is what you want us to focus on. The reflection and refraction. That's what I'm thinking about. Basically, states if you have a surface here, and if you have a light, light, ray of light coming in or sound coming in, vibrations coming in, or billiard ball coming in, right? Because this could be map view, looking at a pool table, right? Then this will bounce at an instant angle, right? It's just physics. And if this is a medium where light is penetrating, then there's, you know, depending on the density of the medium, if this is going down and it's less dense, let's make it more dense. Okay. If it's transparent, it's translucent. I forget what the word is. If this is more dense than that, then the beam will reflect this way. If the bottom is less dense than the top, then it will go this way, right? Let me know if this is what you want us to talk about and what aspect of what you want us to talk about. Let's deal with trig. Here's a triangle. Let's call this angle theta, let's call A and B the legs of a right angle triangle, and let's call C the hypotenuse. Okay. Now, if you do trig now, who was it that wanted to know trig, trig, trig, trig, trig, trig, trig, trig, algebraic functions, adding, subtracting, algebraic functions. Let me write this down here. Maybe we do get to polynomials, algebraic functions, algebraic functions, add, subtract, and subtract. Okay. Just a little notes for myself. Who was it that asked about the trigonometry? I hope you're still here. If you want to know about trig, I've got a whole bunch of videos on trig. There's two different types of trig videos I have up. Okay. A bunch of videos that I did in the late 2000s and 2009, probably 2007, 2000, 2008, 2009, where we did trigonometry. Those are the language of mathematics stuff, right? And that goes into the basic Sokotoa, sine, cosine, tangent functions, and Pythagorean theorem, and stuff like this. If you want more higher level trig, I got an ASMR math series out there where we're going to what trigonometry is really about, which is really talking about trying to analyze a circle, right? Yes, this is what I'm talking about. And also, do we need to use the surface normal? Oh, so this is what we're talking about. Yeah? Okay. Yeah, you do. So basically, this is more physics related than math related. Well, it is math. What is applied mathematics? How do you calculate the length between the earth and a star and a star? You look at the motion, you use triangulation, the angle, and I forget what we did, because I took the stuff at the University of Geophysics, right? But basically, I think you do this. Let me do a little thing. If this is the earth, right, we're here, looking on the stars there, and there, you watch the motion of a star over time, and you measure the angle, and you use trigonometry. Dante, how you doing? How's life? I would have to look it up to do further. Gunner controls? Not yet. We had a couple of people, or yeah, a couple of people showed up that might have gone in that direction, but we talked them down. I think I hope anyway. They went away. They went away. I haven't banned anyone yet. So trigonometry is this here. Let's talk about trig. Oh, hold on. We're going to talk about this. The normal. Yes, this is what I'm talking about. And also, do we need to use the surface normal? Yeah, you need this guy, basically, right? Okay. And the angle of incident, I think you measure it from the normal. Do you not? I'm going to try your YouTube tomorrow. Thank you. It's so useful. Yeah, Lance, and let me know. I'm pretty active on YouTube and on Twitch as well, during the live streams anyway. If you have questions, just post a comment on any video that you have a question on. As long as the YouTube spam filters don't zap things because they do, they take out legit comments. If they notify me, I'll definitely reply. And if I don't reply, send another message. I'll try to reply ASAP. Okay. But I think the the angles are here. This is the incident angle is here. I forget the terminology, by the way. Meromax. The example I have is moving ball that we want to bounce off a ball or other object. Okay, what's that? What's the example? So we've got a ball here that's going in at a certain angle. Is it not? Meromax. So if you have this, usually you have an angle that this guy goes in, and it has to come out at the same angle. Okay, it has to come out at the same angle. So what you end up having to do is calculating this angle and then figuring out that angle, right? Unless there's English spins on the ball, and that totally changes the game. So for example, if you have, let's assume this is a surface, and you have this ball coming in, let's assume the ball has a spin like this, right? So if the ball is spinning like this, and this is what you got to keep in mind when you're playing pool, I used to be pretty, pretty good at pool. I could clean the table, right? I used to be able to go to pool halls or clubs and stuff and drink for free. You play for drinks and drink for free all night, right? So you can give the ball a spin. Let's say you hit it with a cue stick here, right? Give it a spin this way. When the ball hits this surface here, if there is friction involved, and in real life there is friction involved, because the ball is spinning this way, it catches here, and instead of reflecting off the normal at the same angle, what it does, it gives it a spin, and it goes, the angle becomes greater, right? So it really depends what type of problems you're working with. This is very complicated to figure out. Very complicated. I don't even remember how to figure this stuff out when there is more of an angle it is. Is it the remaining motion? What do you mean by remaining motion? What's the remaining motion? Like these types of situations you can also have like this, you could have one ball, ball one and ball two, they come together, they hit each other, and they travel as one unit, and if this is let's say 10, this is 5, the weight of this is, let's say they have the same weight, mass, then this ball is going to travel in this direction if they stick together, like the amount of left over when it hits the surface, like the amount left over when it hits the surface. It is the remaining motion, remaining motion, the amount left over when it hits the surface, the amount left over. So do you have a problem that you can write out for us that you're given? That way I can read the problem and try to figure out what it is that we're talking about, like the amount left over when it hits the surface, the amount left over when it hits the surface. There's going to be, are you talking about absorption where some of the energy is lost into the surface? So if this guy's coming in at let's say 10 and is bouncing back at 8, then you lost 2 in the bounce, and that's usually just heat. I would basically have to know exactly what type of problem we're talking about. Until you, and these guys you could have two different situations, you could have them sticking together going off that way, or you could have them bouncing, you know ball 1 goes this way, ball 2 goes this way, what's their motion and stuff like that, but there's a whole bunch of variables we need to know for this. This is conservation of energy when that happens, and that's usually conservation of energy as well, and conservation of momentum problems. Let's do a trick for a second. Let's finish off the trick. That was our timing. Oh, not bad. Not bad. Let's do a little bit of trick, so we deal with that. Here, take a look at trick. Here's three pins. Let me do this, and bring out the thing so I've got the thing set up properly. So let's assume we have this. Here's a triangle. Triangle means a polygon with three different angles, right? Close polygon. So just imagine if we had this triangle, right? Now I'm going to try not to show all the logos and stuff that they got the logo all over this thing, right? I wish I had pens here, drumsticks here that I've done with before, right? So take a look at this. If I decrease this angle here, right, which side is getting smaller? It's this side, right? If I decrease this angle here, this side is getting smaller. I will search if I have an example that expresses what I mean. Okay, awesome. That'd be great. So the way it works is this, with a triangle, an angle controls the opposite side of a triangle. So this angle, if I draw this triangle, check this out, if I draw this triangle, this angle controls that side, this angle controls that side, and that angle controls that side, okay? Straight up triangle, right? So if you know this, just from this principle, I could give you a triangle and ask you if the triangle is a legitimate triangle. So for example, if I give you this triangle, I say this is 50 degrees, this is 40 degrees, and this is 90 degrees, okay? And one of the other things here's a properties of triangle. Sum of angles in triangle is equal to 180 degrees, okay? The sum of the angles in a triangle when you're quitting in geometry, which is flat surface geometry, has to equal 180 degrees. So if I draw any triangle, here's a whole bunch of triangles. This plus, this plus, this is 180. This plus, this plus, this is 180. That plus, that plus, this is 180. That plus, that plus, that is 180. 180, 180, 180. Sum of the angles in a triangle have to equal 180 degrees. Period. Done. Right? Now, take a look at this. So I drew a triangle here for you, and the sum of the angles is 180 degrees, right? And then I gave you sides for the angles. I said this is six, this is five, this is nine, okay? The principle we have is this. An angle controls the opposite side, right? So this angle controls that side, that angle controls that side, that angle controls this side, the whole side, right? I could ask you a question. Is this triangle a legitimate triangle? Could it be a legitimate triangle? And the answer, obviously I give you one that isn't, so I could prove a point, right? This triangle is not a legitimate triangle. And here's the reason why, right? Sile law, here's the reason why, right? If an angle controls the opposite side, so this angle, this angle controls this side, right? So over here, this angle controls this side, which is the five, this angle controls this side, which is the six, and this angle controls this side, which is the nine. Now, the biggest angle is 90 degrees, and that's across from the largest side, that's legit. And then we've got the smallest angle is across from the side that's six, and the mid-size angle, the one that's between 90 and 40, is 50 degrees, controlling five. This cannot be a legitimate triangle, because if this is the smallest angle, and if it controls the opposite side, this has to be the smallest side, right? So if I make this one a five, and this one a six, and I ask you if this triangle was a legitimate triangle, you would say could be a legitimate triangle. The other one was absolutely not a legitimate triangle, okay? Illegitimate triangle, out of here, right? Is it legitimate? This one could be legitimate now, right? So keeping that in mind, which is an angle controls the opposite side, right? I'm going to erase this guy as well. If I give you a right angle triangle, all of a sudden we have three other equations that pop up here, right? Another equation we have is for right angle triangle, actually we've got four equations, but we'll do them one at a time, right? So this is our first property of a triangle. This is any triangle. The sum of the angles in a triangle have to equal 180 degrees, right? If I call this beta, you know, let's not call it beta because people might get confused with that. Let's call it alpha. No, we don't want to call it alpha. I guess we'll call it alpha. Should we call it alpha? Let's call this w, and let's call that z. Just simplify things so we don't have any problems, right? So if I give you this triangle, it's a right angle triangle. Now for right angle triangles, you have the following four equations you can use. a squared plus b squared is equal to c squared, okay? And that's straight up Pythagorean theorem, which says for any right angle triangle, the two legs of the triangle squared add it up equal to the third leg squared, right? It's pretty common to pair a and alpha, b with beta. Yeah, I put that one theta. No, c with gamma. c with goes with gamma. Racer kill? By the way, high racer kill. I didn't realize c with gamma. But we're just going to call it w and z. It doesn't make a difference, right? The beta I should have put here, that should be alpha and then beta and then gamma. What's the symbol for gamma? Gamma is this. I forget. Pythagorean theorem, a squared plus b squared equals c squared. So if I give you any right angle triangle, as long as you have or angle c opposite of c, yeah, you know what? Let's call it that here. We'll do this. We'll keep it. And I really dislike the convention they use, by the way. I don't personally, it's not my favorite, but basically what they do, they say the capital letters are the angles and the small case letters are the sides, right? Yeah, that's what they do. I'll make the c a little bit smaller so it shows as a small c. Now, when it comes to this situation, and by the way, for any triangle, if you have three of the pieces of information, because there's six unknowns in a triangle, there's three angles and three sides, right? If you have three of the bits of information, one of them has to be a side, you can find everything else. I think small and capital is good, but it's annoying for letters like c. It's very annoying for letters like c. Very annoying because c looks just like c, except for size. I don't like it personally, but it's really, it's the convention that for some reason it's been being used for the last 15 years or so, which is weird, right? So, if I give you any right angle triangle, or any triangle, really, and if I give you that one two five x, w, z, and if I ask you to find this, you can do it. If I give you this triangle, right angle triangle, and I say five seven two, I didn't give you any angles except for the 90. Oh, I don't need to even give you that one. Oh, that's the same as that one. Actually, this doesn't even have to be right angle triangles. You can figure that one out, right? But if I give you this, here, let's call this 50 degrees, and that's 90. You could figure out that stuff. If I give you, kill that, if I give you this, you can figure out the rest. But if I give you this 40, if the sum of the angles and the triangles are 180, then 180 minus 90, because that's 90 is 90 minus 40 makes this 50. If I give you this three, you can't figure out this triangle because you don't have the scale, right? You don't have a distance. You don't have a length, okay? So, as long as there's six bits of information in a triangle, three sides and three angles, as long as you have three of them, one of them being a side, you can figure out all the rest, right? So, for a right angle triangle Pythagorean theorem says this, the legs of the triangle, and the legs are the ones that connect up to give you the 90 degrees, the legs of the triangle, each one squared at it together, is the hypotenuse squared. The other three angles get trialed up to scale, so you know the ratio. Yeah, you know the ratio. That's it. Race to kill. Thank you for clarification. So, check this out. If I give you a triangle, if I say this is 30 degrees, if that's 40 degrees, 30 and 40 is 70 degrees, right? Subtract that from 180, so this would have to be 110 degrees, right? If we have this triangle, the ratios of the sides, we know, right? Because no matter how small or how big this is, the ratios side A, B and C, A over B, will always be the same. Here's another one. And if I said this is the same angles, 30, 40 and 110 degrees, and this is X, Y, Z, then A over B has to equal to X over Y. Z over Y has to equal C over B, right? So, proportions matter as well, and they come in very, very handy. Triangles are very, they've got some pretty cool features to them. They've got pretty cool features to them. Here's the trigonometry aspect of the formulas, right? Sine theta, which is basically, they give you the sine theta is opposite over hypotenuse. Cos theta is equal to adjacent over hypotenuse, and tan theta is equal to opposite over adjacent. Now, what does this mean? This means the following. The theta is the angle where you're at, and relative to the angle, that's what the opposite in hypotenuse and the adjacent are going to be. The hypotenuse is always across from 90 degrees. Right? So, if the theta is here, then sine of B, sine of B is equal to opposite over hypotenuse. So, opposite from B is B over the hypotenuse is C. Cos of B is equal to you put yourself, whenever they're giving you an angle, just imagine yourself putting yourself there, and cosine is adjacent over hypotenuse, right? So, that would be A over C, because that's the adjacent over C. Now, if you put yourself here, this C, the hypotenuse, is also adjacent to B, which is a little confusing when they call it adjacent over hypotenuse. C is also adjacent to B, but C already has a name. It's called the hypotenuse. So, you don't refer to that as the adjacent or the opposite. That's always the hypotenuse. I still don't like this. They're being 360 total degrees. It feels like a product of the imperial system. It's, you mean for a full circle, 360 degrees? It's based on the sun. It's based on, I looked into this a while ago, someone asked this question before and we looked into it and there was a reason for it, which was a pretty legit reason. I forget what it was. Is it based on the sun? So, cos of B is going to be adjacent over hypotenuse. A over C, tan of B, tan of B, is going to be opposite over adjacent. So, you put yourself at B and you go opposite, B over A, B over A. On the same note, one reason is I think it's highly divisible. Would you rather it be 2 pi? It is radiance, 2 pi radiance, right? And 2 pi comes in way more handy than 360 degrees, right? Like radiance is way better than degrees, but degrees because when tan forever would just relate to degrees better, right? For in grade 12, they change it up to or grade 11. If you're lucky you get a teacher, they introduce pi and all of a sudden you're like, what? Yeah, the old mathematician has looked at the sun and figured that the sun took approximately 360 days around us. Is that what it was? That simple as that? Damn, that's a stupid reason to make a circle 360 degrees. But I guess at the time it worked. You didn't have calculator. You got to keep in mind, you didn't have calculators way back then, right? So, you had to simplify things as best as you could. The more you know, the more you know, the more you realize how ridiculous our world is, right? Let's say we wanted to find sine, cos and tan of angle A. So, if you go sine of A, sine of A, sine of an angle is opposite over hypotenuse. So, what you do is you put yourself on A, oops, you put yourself at A and you go, what's opposite from A? From here, the opposite is here and the hypotenuse is here. So, sine of A is A over C. Cos of A, oops, cos of A is B over C and tan of A is A over B. Now, take a look at this. A over C. Where else do we have A over C? A over C. So, what that tells us is cos of B is equal to sine of A, right? These are sort of properties that pop up. Yeah, how are you doing? Welcome, welcome to the live stream. Mathematics, right? B over C. Oh, B over C. So, sine of B is also equal to cos of A, right? This is some stuff that we delved into during our trigonometry playlist, right? While we're dealing with more how they're related to circles and whatnot, right? So, this is what Sokotova represents and the best way to learn is to do a problem. So, let's do a problem. That way you'll see, or a couple of problems, you'll see how this plays out with real numbers, right? Oh, I didn't raise the 10. No, I didn't raise the 10. Cool. So, let's assume we had the following triangle and we wanted to find, we wanted to solve the triangle. Whenever they say solve a triangle, they mean find all six properties of that triangle, okay? So, here's question one. And we're only going to deal with right angle triangles, okay? So, I've given you a right angle triangle. Let's assume I make this a 10 and I make this 60 degrees, okay? I want you to find x, y, and z, okay? Perfect timing. I'm covering exactly the same topic. Now, awesome, Lydia. I hope it helps. Here's an example, right? So, what they would say for this, they would say solve the triangle. Solve this triangle, okay? Apologies about my writing. I write like a scroll, right? So, my question to you is, and whenever you're doing these types of problems, you should ask yourself, what do you want to solve for first? Right now, we have choices. You could solve for any of these first, right? Sometimes, you don't have a choice. Sometimes, you have to solve for one before you can move on to the other two, okay? But right now, ask yourself, which one do we want to solve for first, okay? Have you figured it out? Do you know? A basalt calculator. I've actually made it 60 degrees because that's a special triangle. We know the ratios. We can actually do this without a calculator, okay? It's easy. Z, Z. Very good, right? An alley cat. Because that's the easiest one to figure out. Why is it the easiest one to figure out? Because we have five formulas, right, that we use for right-angle triangles. Two, three, four, five, right? Well, four formulas that only work for right-angle triangles, and one formula that works for all triangles, right? The first formula says the sum of the angles in a triangle is equal to 180 degrees. It doesn't make a difference what type of triangle you have. As long as it's on a flat surface triangle, you can figure it out. It follows this principle, right? So some of the angles, this angle plus that angle plus that angle, has to equal 180 degrees. Now, a lot of people do this. They always add, because this is a right angle, that's 90 degrees. They always add 90 degrees to 60 degrees and then subtract from 180. If you know this is 90 degrees, 180 degrees minus 90 is 90. So whenever you have a right-angle triangle, if it's a right-angle triangle, the sum of the other two angles has to equal 90 degrees. So you don't have to add this to this and then subtract from 180. You just have to go, if this is a 90-degree triangle, then this plus this has to equal 90. So 90 minus 60, z is 30, right? So we figured out z. This is 30 degrees. That was easy. Okay. Now we have a choice. We had a choice before, but because we only got two sides left, our choice is, do we want to solve for x first or y first? Which one do we want to solve first? x or y? x or y? What should we solve first? Difficult choice? Should we choose one? Let's start out with the lowest alphabet, right? x. Let's solve for x first. So if you're going to solve for x first, opposite. Opposite which one? Here's the key, right? If you're going to solve for x, you have all three angles, right? So z was 30. Let me put this here so we know it's this angle here. It's not sitting out in the middle of the triangle. 30 degrees, right? x is easier if you know sine of 30. Yeah, y is the same. It's going to be the same, right? But let's assume we want to solve for x. That's your first question, the decision or your first dilemma. Which one do you want to solve for first? Or which one can you solve for first? And then when you figure out it's x you want to solve for, you're going to ask yourself, do I want to use 60 degrees or do I want to use 30 degrees? Because you can use both of them right now, okay? If you're going to use 30 degrees, you ask yourself, what's x? What's the position of x? It will be half. It will be half. What's the position of x relative to 30? Well, x is opposite from 30. So once you decide you want to solve for this, and then you decide you want to use 30 degrees, take your pen and go, I'm at 30 degrees and I'm going to solve for x. So what's the position of x relative to 30? That's opposite. And what length do I have right now? Well, I have the hypotenuse. So I'm looking for something that has opposite and hypotenuse if I'm going to use 30. Opposite of hypotenuse is sine. So this would be sine of 30 degrees is equal to opposite, opposite x over 10. Okay? And if you cross multiply, like looking at the horizon, like looking at the horizon, you cross multiply this up. So x is equal to 10 sine of 30. And sine of 30 is a special triangle. 30, 16, 90, 1 squared of 3, 2. Sine of 30 is 1 over 2. So this would be 10 times 1 over 2. You can punch in your calculator if you want, which is 5. So you just figured this out. That's 5. Okay? Now, keep in mind, you could use 60 degrees to find x. Here, I'll keep that one up, the special triangle. Yeah, special triangle is just a ratio. It's a set ratio. It works, right? We talk a lot about this in the trick playlist. If this is 30 degrees, that's 60 degrees, that's 90. The ratios of the size are these. So let's assume we're not going to use the 30 degrees. We're going to try to find, let's assume we haven't found this yet. We want to find x, but we're going to use 60 degrees to find x. And personally, I usually always use the information they've given me instead of the information I've calculated to do the next calculation. Just in case I made a mistake calculating this, then the mistake doesn't carry over to the next problem. To the next unknown you need to solve for. So if I'm going to solve for x, I'm going to use the 60 degree angle. So I'm going to put myself at 60 degrees. And I'm going to say, okay, what side do I have? Well, we have the hypotenuse. So I'm looking for, it's going to be one of these guys because they both have hypotenuse and it's not going to be 10 because it doesn't have hypotenuse. So I have the hypotenuse and what's x relative to 60? It's the adjacent relative to 60. So I have to use cosine. So this becomes, cos of 60 degrees is equal to adjacent, adjacent x over 10. Cross multiply, x is 10 cosine of 60. Okay. What's cosine of 60? You put it here, cosine of 60 is adjacent over hypotenuse 1 over 2. So this becomes 10 times 1 over 2, which is equal to 5. This is equal to 5. And you notice cosine of 60 is the same thing as sine of 30. These are properties that show up. Cool? I hope that's okay. I talk like mad in this stream. Craziness, craziness. Fun stuff, though. And trigonometry is ridiculously important. It's really ridiculously important. Once you know how to solve, use Sokotoa, the Pythagin theorem, and some of the angles are 180 degrees, two months of grade 8, 9, you'll ace. Okay. And you're set up for math 11 and math 12, and you begin to have a way better appreciation for what all of this stuff means. And it's ratios. Okay. It's ratios of one side relative to another side. Let me explain to you what that means. Okay. I'm going to erase these. And if you do need the stuff, you can definitely, on the video, we're going to load it on YouTube and it'll be available on Twitch I think for a couple of weeks. You can definitely take screenshots of it. If you learn basics of this, it's so easy to solve all problems. Yeah, basically a lot of problems. All right. So take a look at this. I'm going to erase these guys now. Let's take all of this now. What was the other one? Algebraic functions, adding a subtractive. Oh, those ones we didn't get to those ones. If you're still here, the person that asked for algebraic functions, let me know we'll do speeding on Zalasta. Until then, let me show you where this can be, or some of the other questions you can use. Very useful in real life. Useful if you're going to chop down a tree. Useful in triangulation. Useful useful in chopping down a tree. Useful trying to do circular motion. Useful in cyclic functions. Useful in trying to understand how light waves and sound waves work. Useful in trying to understand how our economic system works, how stocks behave, and the business cycle. It's ridiculously important, right? So take a look at this. Here's our triangle. Here's 90 degrees, right? If I say this is 30 degrees, and here's another triangle. Oops. And I say this is 90 degrees, and I say this is 30 degrees, right? I say this is 5, and this is x, and I say this is 6, and this is y, and I say this is w, and that's z, right? Did I give you too much info? Do I give you too much info? Actually, I even gave you too much info to a certain degree. I think I said algebraic functions, but I mean functions, minus fractions, tomato, tomato, right? To a certain degree. Here's what you can do, right? Here, let me erase this. Let me do it simpler to a certain degree. Let's say I drew this, and I said this is 90 degrees. So what you have right now, because these two triangles, this triangle, right, and this triangle have the same angles in them, they're called proportional, so you could always find out what this side is, because what you would do is, oh, you need one other thing here. You need a total distance here. Let's call this 15, right? What you could do is say, do we need that one? Oh, we need this guy. Hold on, we need this guy. Let's say this guy. We do need, I'm trying to do the speedy ones, so I'm, what do you call it, forgetting a little, little parts of it. No, sundryl. I use it for geolocation, Cartesian coordinates, no? Yeah, yeah. We did a lot in geophysics when we were trying to place ourselves with the data where it was going, right? So if this distance here is in six, and this distance over here is four, we want to find w. If these have the same angles, right, because that's the same angle as this, they're proportional triangles, all you would do is say, oh, five over six, or I would do it this way personally, I go five over w, because they represent the same side, over w has to equal this distance divided by that distance, which is six over 10. And then you just cross multiply and figure out what w is. So 50 is equal to six w divided by six. So w is equal to three, 25, 25 over three. That's the distance here, 25 over three, which is seven and, no, sorry, six and a third, right? Derivation and integrals are fun. Integrals, oh, I missed those from school. Do you miss them? How about a little calculus? Derivatives, also statistics probably, base theorem, all base theorem, look the base theorem forever, 8.333. Oh yeah, 8.333 divided by three, not six, 8.3333. Thank you, Kegan. We've done a little bit of stats. We haven't really done calculus. I'm not really touching calculus right now. When it comes to algebraic functions, if I'm thinking about it the right way, functions minus fractions. So let's assume we have something like this, derivative of a maximum. I said base theorem for a semester. Oh, it's crazy. I haven't looked at base theorem. I don't even remember what it is. When I was in grade four, I was struggling with multiplication. I had a subteacher once and she taught me lattice multiplication, which instantly stuck and I used it to this day. Is there any other methods to work out long division that you know of? Because long division still doesn't stick, even if you just drop the method. I can Google it. You know what? I just use long division. If I'm using numbers, if you're using polynomials, if you're dividing polynomials, use synthetic division. But long division, the only reason it didn't stick is because maybe it didn't teach it to you properly. I checked on your YouTube and subscribed. I'm glad I found your stream. Looking forward to future streams. Have a good night. Good night, Lance. Thanks for popping by. Thanks for the sub. And we're going to do at least two of these math streams a month. So far, we've been doing one a week, really. Okay. Look at that functions. Young Jeffrey. Twist mic. How are we doing? Here's algebraic functions, algebraic expressions, if you want to deal with them. 2x minus 6 over 5 minus 3x plus 2 over, I don't know, 6. Is this the type of stuff we're talking about? How's it going? Good. Good. We're doing mathematics. So life is sweet. Mad, mad steering. Thank you for the bits, brother or sister, of course. Thank you. Good night. Good night, Matt. Good night. Cheers, reward to three others in chat. Nice. Nice. Nice. Thank you for that. What's interesting is the enormous theorem. Yeah, in high school, I struggled with those. Now, take a look at this. This is just adding subtracting fractions. And this is something that takes people out of the math game. Really. It's one of the first places that people struggle with. Like, I've done this with students for the last two decades, right? If I find them, you know, they're not paying attention, or they're struggling with stuff, and they need a little booster, they need a little engagement. I need to explain to them what math is, which is just that language, right? I always ask students, what's the one thing you hate about mathematics, right? More than 50% of the time, because I'm dealing with high school kids, they say, I hate fractions, right? So for me, as soon as they say I hate fractions, Mr. What Bitcoin? Yeah, this is where I usually drop out. Yeah. When they say, tell your students to solve this, you need three equations for that. So whenever they tell me they hate fractions, I go, okay, so if you don't like fractions, that means you can never eat half an apple, because a half an apple is a fraction, it's half of an apple. That means you're going to go walk around your life, your whole existence, dealing with just whole numbers, right? And they're like, what? And I expand this a little bit further, right? So fractions is basically, name the game is, and adding the subtracting fractions, by the way, it's harder than multiplying and dividing fractions. Multiplying divided is easier than adding subtracting. Do three little problems just to get them. So when you're doing these types of things, name the game is, you need a common denominator. And if you've already done the first introduction to fractions, the common denominator for this is, and by the way, if you guys are struggling with fractions, let me know, and I'll show you how to deal with fractions in the next stream. Okay, I'll lay it out for you the way I teach you from the base down, or base up, because of common denominator. Common denominator. What is the common denominator? Common denominator, you extract out through breaking these numbers down their prime factors. You're looking for the smallest number that both five and six divide into. And this is when you do it. LCD. Let's go. Yeah, the lowest common denominator, right? You can also refer to it. Instead of a third, they like that with a three that goes to infinity. It totally makes sense. Yeah, ridiculous, right? Crazy because they give calculators to kids in elementary school. So they know how they know how to be a monkey and push numbers into a calculator. And half the time they push and punch it in wrong. And they don't have no idea how to deal with fraction, which is insane. Right? Here, let me show you how to do this. And then I'll kick it back a notch. And I'll show you how you can find the common denominator. Okay, while we're at this right now. So common denominator for this would be 30. What's the smallest number that about five and six go into evenly? Right? 30. You ask yourself, what did you multiply five by to give you 30? You multiply five by six? And you, by the way, the common denominator could always be both of these things multiplied together. But you don't want to do that because the numbers could become extremely large. If you had a smaller common denominator, better, right? Smaller numbers are easier to deal with than larger numbers. So you multiply this whole thing by six. So this becomes 12x minus 36. And then you got a minus. Whenever you have a minus, put everything else after the minus sign in brackets. I'm lazy. I just always use the product that don't do it dice power. It becomes way harder. I'll show you after this one. What did you multiply six by to give you 30? You multiply by five. So five multiplies everything here. Now, when you have a minus sign, again, put everything in brackets. So five times that is 15x plus 10. And then what you do is this negative sign affects both of those. So this becomes 30. That's 12x minus 36 minus 15x minus 10. And then you just combine it like terms, right? So 12x minus 15x is negative 3x. 36 minus 10 is negative 40. 46 divided by 30. That's it. My teacher used to push me in school because I was able to do the work in my head. I explained that the extra steps felt redundant to me and wasted time. The only limited resource we actually possess. I understand the logic behind showing this work to measure our logic process. I just received a negative reaction. Yeah, that's unfortunate. But the kicker with doing stuff in your head, right? When you're first being introduced to a mathematical concept is because when you're first being introduced to a mathematical concept, the problems are easy, right? But the method applies to even the hardest problems. So if you are doing everything in your head for easy problems, most people really don't understand how to do them for hard problems because you can't do them in your head. So you have to know what the process is because they think you're copying someone. No, it's not because of copying. Some maybe, but it's not because of copying. I argue it's easier when your denominators are functions of one or more variables. I don't know if it's easier. So the work allows the reduction of fault in a harder concept. Yes, basically. So take a look at this. Let's assume we have this here. I'll show you how you calculate the common denominator for numbers. And if you guys want later on in the next stream, we'll do rational functions. Let's assume we have the following. 1 over 24 plus 3 over 70 minus 1 over 28. What's the common denominator? If you multiply this times this times this, it's going to be a gigantic number. It's okay because then these are just numbers. But what if there was plus x minus w and stuff like this in there? This is the way you do it. Break these down into the factor trees. So 24 breaks down into 2 times 12, 2 times 6, 2 times 3. 70 breaks down to 7 times 10, 2 times 5. 28 breaks down into 2 times 14, 2 times 7. Here's the way the lowest common denominator works. Take the biggest grouping of any individual number and multiply them together and that's the common denominator. So if you're going to add all these guys up, you look at the first breakdown first. And anything you broke down, whatever you broke down is already gone. So this number is 24, which is 2 times 2 times 2 times 3, right? The answer is this. You did it with a calculator. So take a look at this. The biggest grouping of any individual number. There's three 2s here, one 2 here, two 2s here. So the biggest grouping we have is three 2s. So the lowest common denominator is going to be 2 times 2 times 2. And then we look at the next number. Here's a 3. One 3, no 3s, no 3s. So the biggest grouping of 3s we have is one 3. I'm done for my mind. So you got times 3. This one, you got a 7. You got one 7 here, no 7s here, one 7 there. So you need a 7 times 7. You're done with all these numbers. Oh, you got a 5 here, no 5 here, no 5 here. So you need a 5. The common denominator is this. Multiply them all out. That's your common denominator. Now what are you going to multiply this guy by? Well, take a look at this. This guy already has 3 2s and a 3. Here's 3 2s and a 3. So you multiply the 1 by 35 because you already got those ones. What do you multiply the 3 by? Well, you have a 7. You don't need that. You got a 5. You don't need that. And you have a 2. So these guys are gone. So you multiply it by 4 times 2 is 12. So you multiply 3 by 12. 28. You got two 2s and a 7. Two 2s and a 7 you already got. So you multiply the 1 up there by 6 times 5 is 30. So you multiply that by 30. Right? Powerful, powerful, powerful, powerful, powerful. You find the largest common factor without prime factorization. This is especially important for large numbers, which are hard to factorize. Yeah. But for small numbers, prime factorizing easier by hand. Yeah, it is for sure. My mom makes me watch this channel just smoking. Fun guys, fun. Thanks for being here. We'll call it a stream. It's been a couple hours. And we went full on today. Fantastic. That's what I love. No one's full on. We did physics, we did mathematics. Thank you for the questions, by the way. Mask of Raven, thank you for the help. Racer killed, thank you for the help. Thank you for the clarifications. Thank you for correcting any mistakes I made. Thank you for the subs. Thank you for the follows. Thank you for the bits. Right? Great stream. Very fun. Very fun. How often do you stream? I stream two to three times a week. That's what I do while I'm at. Great stream. You're all great stream. Fantastic stream. And because I have students and I do other things and I'm shooting videos and stuff like this, I haven't set up a regular schedule. They tell me to be successful on Twitch. You need to set up a regular schedule that people can know that they're tuning in at that time. I'm not there yet, I guess. I'll add that in when we're going to kick things up a bit. But right now this is working for me. This is, you know, it's working for a lot of people that are here. And I do announce the streams on our Discord page, like two to three days beforehand. I announce them on Twitter. I announce them on Gab. I announce them on Mines. And I announce them, announce the streams on my Patreon page. Right? So there's notifications that go out if you sub to me. You know, you don't even have to donate money on Patreon. You can just follow and send out the notifications. Right? If you can support it through Patreon, fantastic. That's great. If you can sub through here, fantastic. Right? But it does, I do announce the streams two to three days beforehand that we do it. Okay? That is the most important thing. That is, it has gotten at the end, but very enjoyable. Thanks. This strain. Thank you for being here. I'm glad you made the last little part. I was going a little fast. Speed goes out of style, but I went in speed mode now because we're trying to cover as much as we can. We didn't fit it all in. Right? Wish there was more teaching, learning channels on Twitch. I think it's coming. I think it's coming. Just to let you know, like a few months ago, I had someone from Twitch contact me saying that we've noticed you're doing things other than gaming. We'd like to talk to you. Right? Oh, JJT, you missed the stream. I'll be doing that. We're doing one tomorrow. Talking about relationships. Starting at, I don't know what time we're starting. What time are we starting tomorrow? What time are we starting tomorrow? What time are we starting tomorrow? Tomorrow, we're starting at, what are we calling the stream tomorrow? Open discussion on relationships. News and don'ts of human interaction. We're starting at 3 p.m. tomorrow, 3 p.m. my time, PDT, West Coast Canada, United States. Thank you very much for the stream again. As always, it is a privilege to be able to access this space you create for us. My pleasure, really, my pleasure, man. I hope you guys have a fantastic weekend. And if you can make it tomorrow, I'll see you guys tomorrow. And then later on the week, I'll announce the next streams for next weekend. That's it for now, gang. Thanks for being here. Fun, fun stream. Bye for now.