 In lecture one of this series, we talked about the idea of distance, where distance represents how much space, how much length is between two objects, right? And we use the distance formula to calculate that. That is a form of distance measurement, right? We can measure the distance, which might be called length, or width, or height. With those, the difference between those words is somewhat arbitrary. That kind of has to do with direction. Length and width usually represent some type of horizontal thing. Height often represents a vertical distance. Again, those are kind of just words that have sort of colloquial meanings, but they all represent distances. And how can we measure distance? Well, it turns out there's a lot of different ways of measuring distance. You can measure distance in feet. You can measure distance in inches. You can measure distance in miles, just to name a few. And it depends on what you're measuring to decide what you're going to do, right? If it's a very, very large distance, you might be measuring it in miles, like the distance between countries or cities or states. If you're measuring the distance on a more moderate scale, maybe you're measuring in feet, right? It's like, oh, these things are 200 feet away from each other. Or if things are relatively close, you might be using inches, or you might be using the idea of a yard, for example. And there's these different measurements of distance you one could use. Like for example, I had my son told me a joke earlier today that how do you measure a snake? How do you measure the length of a snake? And I was like, I don't know, how do you measure the length of the snake? And he's like, oh, you have to measure the length of a snake in inches because they don't have any feet. And so when it comes to measurements, it's really, it has a lot to do with relativness, like how close are they together. But there's also a lot of culture that's built into measurements. Because when it comes to feet, inches, yard, miles, why not use things like centimeters, or meters, or kilometers, or nanometers, or we can keep on going, right? There are some competing forces of competing measurement systems one could use. And so for those, for those people, generally speaking, who were raised in the United States, if you're measuring distance, you're quite comfortable with sort of this traditional British method. But many places in the world, with the exception of the United States, have migrated to more the SI units, SI units, which, because there's some benefits to them in scientific measurements, mostly the switch between the different measurements is much easier, because it's usually multiples of 10 or multiples of 1000 or things like that. And so for some people, if you're like, oh yeah, my house is five kilometers away. Some people have no idea how large five kilometers are. But if you're like, oh, it's like, my house is three miles away. I'm not saying those are equivalent to each other. But some people might have that frame of reference right there. And so measurements, the measurements we try to use have a lot to do with culture. And sometimes there are measurements that are introduced that have some more scientific advantage. But still a lot of it has to do with culture. And so some measurements might feel natural or standard or straightforward. But that's, you know, other ones might feel awkward or bizarre or unnatural. But those are really just our perceptions. There's nothing necessarily better about one or the other in terms of this is a better measurement foot, feet versus meters or something like that, right? Again, that has a lot to do with culture. The reason I talk about this is that angle measurement is the same basic idea. In this trigonometry series, there's two particular forms of angle measure that we're going to focus on. In this video, we're going to talk about the idea of degree measure. And sometime later on in this lecture series, we'll introduce a competing notion of angle measure called the radiant measure. And so let's first talk about the degree measure for a moment. At least in the United States, I would say that the degree measure is the most common, most comfortable, so to speak, most natural way to measure angles. But I want you to be aware that the choice of 360 degrees is somewhat of an arbitrary choice. But this is how we define angle measure. So we want to put a quantity to describe how big one angle is compared to another. And so in the degree measure, a degree is denoted by the drawing a little circle as a superscript here. Same thing like when we use temperature degrees, although it has a different meaning, same notation here. 360 degrees represents one complete rotation. So if you took, take a ray and rotate around in a whole circle, so that is the initial side and the terminal side are exactly the same thing. This represents a 360 degree measure. So there's 360 degrees in one complete rotation. So what this tells you is if there's 360 degrees in a full rotation, a half angle, which is half of a rotation, it's just a half plane, that would be half of 360 degrees. And that's going to be 180 degrees. A right angle, which is half of a flat angle, then, would then take half of that. A right angle would be considered 90 degrees. And that's a statement many of us are probably familiar with. A null angle, which would take no rotation whatsoever, you just start and stop at the same spot, that would then be zero degrees. And so some of these important angles we talked about earlier can be described using a number. How big is the angle? In all, it's based upon this observation that we're deciding that 360 degrees gives us one complete rotation. Some other vocabulary we should connect to that an angle is acute if it's between zero degrees and 90 degrees. An angle is obtuse if its angle measure is between 90 degrees and 180 degrees. And in particular, one could say that one degree is worth one 360th of a rotation. Why 360? What's so significant about that? Well, the 360 degree system, it actually dates back to the Babylonian culture, which existed roughly 4,000 years ago from the recording of this video. This is the same culture that introduced at least US American culture to the idea of 60 seconds in a minute. You have 60 minutes in an hour. 60 seconds equals one minute. 60 minutes equals one hour. And so they have 360, which is six times 60, of course, in a complete rotation. So hence, degrees are deeply embedded into many of our cultures. Even if it wasn't there historically, often it's just so used international, many of us adopted it because of that. Again, it's just so ancient and so heavily used that it feels natural because degree measure is essentially our native language when it comes to angle measurement. That doesn't mean it's necessarily the best or the easiest to use. It's just the one we have the most experience with. And so that's why it feels so familiar to most people. Another thing we should mention about degree measure is the difference between clockwise and counterclockwise rotation. When it comes to a degree measure, if we have our initial side and our terminal side, so our initial side right here, IS, our terminal side TS right here, if you take a counterclockwise rotation, this is considered a positive angle measurement. On the other hand, if you take your initial side here, and you rotate it here to your terminal side, and you take a clockwise rotation, that's considered a negative angle measure. So positive degrees, it's going to be counterclockwise. Negative degrees is going to be considered a clockwise rotation. Why is clockwise negative and counterclockwise positive? Well, again, clockwise versus counterclockwise, this is all just cultural things. There's no reason to think that clockwise is the preferred direction that a clock should spin. After all, clocks, which historically were circles, right? So as this is this our hand goes around, right? Why does it go that direction? Why not the other one? Well, the reason for this really just comes down to sundials. So you have a traditional sundial, right, which is a circle, which has a big triangle on it, which then cast a shadow on the sundial based upon the location of the sun. Well, because of the rotation of the earth, the shadow of the sundial would move around in a clockwise manner. And so that's why clocks were then later, when they were given gears and gears and, you know, cogs and such, they were designed to go clockwise because they mimic the sundial's direction. So it's just because of the rotation of the earth that we got clockwise. What if we run a different planet and it's spun the other direction? And so sundials move counterclockwise. There's no, there's no big importance there whatsoever. So again, well, okay, if it's arbitrary, why would we disagree with the culture here? Well, because of the way we orient the x, y plane, that is, we have our x-axis here and the positive direction is considered right and the y-axis, its positive direction is considered up. It turns out that angle measure will make a lot more sense moving counterclockwise than clockwise. So the way that we orient coordinates in the x, y plane is typically in the following manner. We draw a horizontal line and we denote this as the x-axis. We draw a vertical line and we don't know that as the y-axis. The intersection of the x and y-axis is commonly referred to as the origin as to, in a manner of saying that all points originate from the, this point, the intersection of these two things. So if you have a point in the plane, you can count how far along the x-axis is it, that gives you your x-coordinate and how far along the y-axis is it, that gives you your y-coordinate. That's how coordinates work in general, okay? So that gives us the standard coordinates of a point in the plane. We can also talk about the standard position of an angle because angle is determined by any three points in the plane, right? We just connect them with the rays. But as the measure of the angle doesn't change if we rotate the whole angle, something like this, the exact location of the points don't matter if we're just interested in the angle. So we can put an angle in so-called standard position if its initial side coincides with the positive x-axis, which is this part over here. The right side is considered the positive side. And then the terminal side, well, it just terminates wherever it wants to, okay? And so an angle in standard position would be something like this. Its initial side always coincides with the positive x-axis if you're in standard position. And then the terminal side is whatever it is. And so there's some point over here, x comma y on the terminal side. So if you give me a point other than the origin, that'll determine a unique angle in standard position. And so moving forward, we'll often do that. We'll describe angles via a point on the terminal side, a so-called terminal point that determines the angle, okay? So another thing we should mention about these coordinate systems, the idea of a quadrant. A quadrant, as the name suggests, is just one fourth of the plane. The first quadrant is going to be everything above the x-axis and to the right of the y-axis. And then going in a counterclockwise rotation here, the second quadrant is going to be above the x-axis to the left of the y-axis. Things in the third quadrant are exactly those which are below the x-axis and to the left of the y-axis. And then the fourth quadrant are all of those points which are below the x-axis and to the right of the y-axis here. And so we get these four quadrants. And so the angles that determine the four quadrants, the so-called quadrental angles, are going to be important to us. So the x-axis coincides with the angle zero degrees because the x-axis, if your initial side is the x-axis and your terminal side is the x-axis, then of course that's a null angle that's going to give the x-axis. That separates the first quadrant from the fourth quadrant. The first and second quadrant are separated by the angle of 90 degrees. The second and third quadrant are separated by the angle 180 degrees. Notice that 90 degrees is a right angle, 180 is a flat angle. The angle that separates the third and fourth is going to be the angle measure 270 degrees. And if we were to go one more, we're going to end up with 360 degrees. Well, 360 degrees and zero degrees actually represent the same angle in a manner of speaking. It's not the same angle measure, but it's the same angle. So what's the difference? Because in terms of an angle, an angle is determined by two rays. So you have your initial side and you have your terminal side. That's what determines the angle. In terms of the measure, we're trying to think of how much space is spanned. Are we taking this angle right here? Are we taking this angle like right here, right, where you do a complete rotation plus a little bit? Or what about this angle right here where we do two complete rotations in a little bit, right? Those all in terms of their initial and terminal sides are the same, but the angle measures are different. This gives us an example of what we call coterminal angles, coterminal angles. Coterminal angles are going to be those angles which have the same initial side in the same terminal side, although their angle measure could be different. For example, zero and 360 degrees are coterminal angles, as observed right here. Let me give you another example. Take, for example, 60 degrees. A 60 degree angle is coterminal to 420 degrees. That is, if I put it in standard position with the positive x-axis, a 60 degree angle would look something like this. A 420 degree angle would be exactly this thing right here. In fact, we know that two angles are coterminal. If their angle measures, if you take the difference, is a multiple of 360. So notice that 420 minus 60 is 360. So that shows you that a 60 degree angle is coterminal to 420, because their difference is 306. That means they're only off by one complete rotation. It's also true that 780 is coterminal to 60. Notice that 780 minus 60 is 720. 720 is three times three, or excuse me, is two times 360. So a 780 degree measure means you go around twice and then you go 60 more degrees past two complete rotations. You'll also notice that negative 300 degrees is coterminal to 60 degrees, because if I take negative 300 degrees and I subtract from it negative 60 degrees, you get negative 360 degrees, which is negative one times 360 degrees. So the difference there would be that starting at the initial side, you go clockwise 300 degrees until you get the terminal side. And so negative 300 degrees is likewise coterminal to 60 degrees. For the most part, there's no reason to distinguish between 908 degrees versus, say, 188 degrees. And there's really no reason to distinguish between negative 800 degrees with 280, because these are all coterminal with one another. There are a few exceptions to that, which I will point out along the way. But for the most part, there will be no difference in terms of measuring different coterminal angles. And so because of that, we will choose our angles to be between 360 degrees and zero degrees. It should also mention that sometimes we need an angle measure that's less than a single degree, because after all, the degree measure puts 360 degrees inside of a rotation. But what if we need something more refined than 360 degrees? In that situation, there's sort of two common ways people use to subdivide a degree. One approach is just to use sort of like a decimal portion, decimal portion of that. So for example, you might get something like 15.34 degrees. And so just to make sure 15 degrees plus 34 100th of a degree. And so sometimes to be more precise, you might need something like that. Another approach that was used more historically, it's not used as much nowadays, but still there are some that still use it is the idea of like a minute. That is a minute is 160th of a degree. And so you might say something like, Oh, I'm going to take 15 degrees and 20 minutes. You put a little apostrophe there to indicate a minute. But how do you what do you do if you get more precise than a minute? Well, one minute is 160th of the degree, right? So one minute equals 160th of a degree where here minutes are measuring, measuring angles not time. You can also subdivide a minute in the usual way that if you take a second, this is going to go 160th of a minute. In which case, then more precisely be like, Oh, 15 degrees, 20 minutes, 24 seconds. And so that gives you pretty some pretty good precision, right? You get up to including minutes and seconds, you can get up to one out of 300, the 3600th of a of a one degree, which is already pretty small itself. So there's very not often where you need to be more refined than minutes and seconds. And so that's a standard unit that is used over and over again. This is kind of similar to like, if you pull out a ruler or measuring tape, if you have like an inch, right, it might be subdivided into a half inch, and then like a quarter inch, and then like an eighth of an inch, and then you have like, what do I say, like a sixteenth of an inch, I keep on going or a 32nd of an inch, you don't typically see very many measuring tapes that have more than 30 seconds of an inch. That's pretty refined. There's not many measurements that require you to get smaller than a one 32nd of an inch. The same thing is also with degree measure. Now for this lecture series, I'm going to exclusively be using just decimal representations. So if we don't get a precise degree, we're just gonna, we'll just route, we'll just do a decimal. But be aware that in some angle measurements, they still use this idea of minutes and seconds.