 Okay, let's continue with our conversation with trigonometry, right? And we've done a fair bit of tricks so far, sort of laid out the ground, the foundation that we need to build on, right? We talked about trigonometry and what trigonometry is, which is basically, or why we study trigonometry. One of the main reasons we study trigonometry is because triangles, specifically right angle triangles, they're related to circles, the perfect cyclic function, right? And so we sort of did a little intro to that. We drew a circle and we called the radius, you know, we measured off nine squares here. You know, the radius ended up being nine and we talked about how we can fit a triangle as a coordinate system and move around the triangle, right? From there what we did, we introduced the concept of the trig ratios, which is basically taking a right angle triangle inside a circle and looking at the ratio of the sides compared to each other, right? With sine being opposite over hypotenuse, cos being adjacent over hypotenuse and ten being opposite over adjacent, right? Sokotoa. And we sort of looked at those ratios and gave them special names, which were sine, cos, sine, and tangent. And what we did with our circle, we took the circle, instead of saying it's, you know, the radius is nine squares, we just standardized that and called the radius one unit. And we just ended up calling that special circle, any circle with a radius of one unit to be the unit circle. And we choose the number one because one is easily scalable. We can, you know, simplifies our calculations, right? And what we did from there, taking the unit circle, we ended up graphing the sine function, the cos function, and the tan function, right? And sine and cos come up to beautiful curves. Basically, any type of wave that you see like that is either a sine, cos, or a combination of both, right? And we looked at the tan function, which is, again, it is smooth, but it has a couple of asymptotes to it, right? And that was sort of our introduction to the unit circle, trigonometry, and graphing the trig functions, right, sine, cosine, and tangent. And we did all those three videos, those three lessons, with the angles being in degrees, right? And degrees is a unit of angle measure that's used early on, but then once you go further and further into mathematics, you start using less and less degree units of measure, and you go into radians. So what we did in the following video, after the first three and the fourth video, I guess, we introduced the concept of radians. And radians is basically us trying to simplify calculations where we took a circle and said, instead of measuring the angle going like this, right, counterclockwise in degrees, we're going to measure this angle relative to the radius of the circle. And what we did, we defined one radian to be the same distance along the circle and the perimeter of the circle that you travel, which is equivalent to the radius. So if this was five units, right, if the radius was five meters or feet, let's say, if you travel along the circle, five meters or feet, whatever the radius is, we call that angle to be one radian, right, it simplifies our calculations. Again, it's just like making the unit circle, right, we want to simplify a calculation. So we reduce the ideal circle, the circle would compare everything to really, to something with a radius of one and our angle measurement for radians is something that eliminates our need to understand what degrees is to work with a different unit and we standardize it and we kick it down to one relative to the radius. It's pretty cool, actually, once you see what you can do with just a simple adjustment to your analysis or your interpretation or working with a certain system and simplifying it for yourself, right, simplifying the calculations for yourself. And what we did from there after we took a look at how we can convert between radians and degrees and what radians are and how we can do the measurements, we took a look at how we can measure, you know, the arc length of a circle, if you're traveling certain radians or degrees around the circle. So we looked at how we can calculate the arc length, the distance from where you started to where you went, we calculated the sector area, right. And again, we took a look at how you convert between degrees and radians. So that's where we are right now, if you missed the initial five videos. And if you did watch those, they should have been a pretty quick review and sort of clicking for you and sort of reminding you what we've done so far. Because in mathematics it's really important, since everything builds on itself, right, or within a certain group of genre of mathematics in high school, it's all connected, right. Almost everything you learn in high school is connected, it builds on each other, right. So it's really good to sort of do a review before you continue on with the concept that requires more than five minutes of concentration, right. And this is a review. So if you are just sort of a hint or just a recommendation, if you are going to study mathematics, one of the best ways to learn math is review. Just a little bit of review before you walk into your next class or you watch your next video. It doesn't have to be the whole thing. It's just, you watch a little bit or you look at the formulas and just flash cards that you might have. And this is amazing if you write tests, if you have to write tests, this is a great way to study for this and I will cover this stuff later. But just as a hint, before you walk into a test, five minutes, ten minutes before you walk into a test, flash things that the test is on. And that's going to start getting the juices flowing, sort of a warm up, right. It's like going into, you know, if you play sports, you don't walk on to the court and play the game right away, you warm up a little bit, right. So this is a nice little warm up. Always, always try to keep this in mind, always try to do a little review because you end up learning mathematics way faster, right. You know, just five minutes, two minutes, ten minutes for really long concepts. So that's where we are right now for trig. Now what we're going to do, we're going to continue with our analysis of the circle. And the unit circle and other circles. So what we're going to do, we have a circle here and this circle could be, you know, for us it's really important to take the circle as a representation of the ideal cyclic function, right. And we analyze the circle as an ideal cyclic function that way we, you know, we can understand how cyclic functions work. But you don't have to think of the circle like that. You can think of it as a piece of pie or pizza, whatever you want, right. The circle can be anything, anything you want it to be, right. So if we take a circle, right, if we take whatever system that we're looking at, there are things we want to do to understand the system, right. And one of the things we end up doing with any system really is breaking that system down into equal segments, right. Like if we have, if you have a, if you buy a piece of pizza or a pie, if you've got a, you know, a few people in the room, you want to break that thing into even pieces so everybody gets a piece, right. We also do that when analyzing functions, when analyzing systems. We like to take a whole system and break it down and see what happens in certain segments, right. If something takes a year, sometimes you break it down to 12 parts, right. Take a look at per month basis, right. If you're doing finances, economics, you take anything that's like a year maybe, right. And you break into quarters, right, have quarterly reports for us. For us, for a system in economics, it, you know, could be something based on some type of corporation reporting, right. They're reporting quarters, so you would go, you know, one quarter, two quarters, three quarters, four quarters, right. And 12 months would be the circle broken down into 12 pieces, right. So really important to keep this in mind, when we're studying trigonometry, we're not just studying triangles. We're not just studying circles, some random, random equations and stuff. We're studying a system. And what we're doing with that, we're breaking the system down and seeing what happens to the system, right. So what we're going to do with this circle right now is it's already broken into four quadrants, right. We've broken into four equal pieces. What we're going to do now is break it, break each quadrant into halves, right. So we're going to have instead of four pieces, we're going to have eight pieces. We're going to have one, two, three, four, five, six, seven, eight, all right. So we're going to have eight even pieces. And the other thing we're going to do, we're going to take this circle and each quadrant and break down the quadrant into three parts, right. So right now we have 90 degrees. We're going to break it down into 45 degrees and 45 degrees. And we're going to take 90 degrees and break it down into 30 degrees, 30 degrees, 30 degrees, right. You can see where this goes. This is one part. We're going to break it into two parts. We're going to break it into three parts. We can break it into four parts if we want, five parts if we want. It's many parts as we'd like, right. So that's what we're going to do right now. And since circles, the system, the core, the foundation of it, when we're talking about the coordinate system, what we're going to do is we're going to base it on the coordinate system and again do triangles, right angle triangles into this. And there's two special triangles you have to learn. These two special triangles and they come up all the time in high school mathematics anyway. But what we talked about is one of the reasons they show up and one of the reasons our schools get everyone to memorize these things is because they're, they allow us to, you know, delve a little deeper into circles, into systems, right. So the two triangles are 45 degrees here. And automatically if there's going to be a right angle triangle, that's 45 degrees, this is 90 degrees. And some of the angles in a triangle have to be 180 degrees. So this guy ends up being 45 degrees. So one triangle we have is 45, 45, 90, right. Another triangle we have, if we're going to break 90 degrees into three pieces, we need each piece, we're going to break it down to 30 degree pieces, right. 30 times 3 is 90 degrees. So another triangle we have, we're going to have one here which is 30 and this is 90, so it makes this angle here 60, right. So 30, 60, 90. And if we go another 30 degrees, it means this is 60. Well, if this is 60, it goes up here, that becomes 90, so that becomes 30. So 30, 60, 90 is the same thing as 60, 30, 90, which is beautiful, right. So instead of having to learn three triangles, we just have to learn two special triangles. And that's what we're going to do right now. We're going to create the triangles here and see how these triangles are related to our circle. And we're going to redraw the circle because we want a clean circle to start off with, okay. So we'll do a little setup, create a circle here and we'll do our calculations here. We'll talk about what the special triangles are and how they relate to this.