Hi everyone, and welcome to our general review of periodic functions. Just so you know where this is going, periodic function means things that move in a period or a cycle. There are many things that sort of repeat on themselves. And if you're looking to model many, many things that have a repeating pattern, these are going to be the functions that you want to use. Think of seasons, days of the week, years, cycles, whatever. These are important things to know, and it all starts with studying angles, so let's do a little recall. And we're going to look at two rays, you gotta go in the Wayback Machine and kind of remember stuff from geometry, probably a long time ago. But you have two rays, and if you take two rays, remember, these are little line segments. They originate at a common point, further reminder, this is called the vertex. We have two rays that originate at a common point called a vertex. We denote one ray to be what's called the initial side. It does not matter which ray you pick, usually, we pick the one that's at the bottom of the screen normally, just to make it easy, but you certainly don't have to. And then you imagine a little angle, the second one growing out of the first and rotating in such a way that it stops. And now you have the terminal side, and now you can distinguish between the two rays, and they create what's called an angle. It's pretty common to move in the counterclockwise motion at first, so we're moving counterclockwise. This will change later, but when you're first starting, counterclockwise is the way to go. And you could think of sort of learning to know, how much rotation do I have? How much rotation was needed to move one ray from the first, to create the terminal side from the initial side? And the measure of this angle is usually denoted by the Greek letter theta. So let's put this over here on the side here, so this is theta. And what it's going to measure is rotation, okay? Now it's very important to realize that there's a little confusion sometimes, if I just draw two rays, so this is my caution. It should be pretty clear what angle we're asking for, but if you're ever unsure, make sure your work is clear and make sure you're talking about things clearly. Because for example, if I just draw two rays that meet at a vertex, right, I could mean the little angle that goes between them, that's option one. Sure, and most people, I think will assume that by default. If I draw the same pictures, so I draw a straight horizontal ray and then one up a little bit. What if I want a lap around and then measure it again, right? So think of it like hands on a clock, just because you have two hands, maybe you went around once, you want to know how many times it rotated, maybe you don't. We're also making an assumption here, so same exact picture. So imagine the hands at, I don't know, one at 3 o'clock and one at 1 o'clock, something like that. In the last few drawings that I went, I went in the counterclockwise direction. There's nothing really stopping me from going in the clockwise direction as well and ask, what is that rotation? So one picture, you can imagine, can lead to lots and lots of examples of what angle we're trying to measure. So just be sure that that is very clear in your picture or what you're trying to measure. For historical reasons, we have a common measurement. For historical reasons, actually going back to the ancient Babylonians when they were first looking at this, and they did things base 60, that was just what they did. And so we define one lap around the circle to be 360 degrees. This is completely arbitrary, this is just historical reasons why we define that one full lap is 360 degrees. I think noticed that a rotation is 360 degrees, but they may not be sure like why that is. You could pick any number you want if you were back in ancient Babylonia, doing your thing, doing math back in the day and pick a number. But they did things 360, and that is why today we do things to 360 degrees. It's a pretty fascinating history there, but the good news is that it's common and people know it. And then of course, if you do jumps and tricks or watch skateboarders or the Olympics or something that, they have 180 degrees, and you can break up the circle into these sort of things. And then you have a quarter of a circle is 90, it has nice numbers to it. Unfortunately, though, this isn't usually the measure that we like to use when we do some sort of advanced math. You see this all over science, where there's Fahrenheit and there's Celsius, but scientists use Kelvin, some other one that just kind of makes sense when you're working in. So we're going to use the scientific way, the mathematical way, the way that kind of works out better for your calculations, and this is called radians. Now, when you see radius, just realize it's another system to measure angles, just like Fahrenheit and Celsius are two different systems to measure temperature. So we define one lap around, so 360 degrees is going to be 2 pi radians. So if you want to know where the 2 pi comes from, you have to go back and think of, what is the formula for the circumference of a circle? Circumference of a circle, maybe you remember it, maybe you don't, it's 2 pi r. So when we talk about circles and we just need a circle to talk about, we tend to pick the unit circle, which is a circle of radius 1. So r = 1, and then 2 pi r just becomes 2 pi. 360 really is sort of humans trying to put their number system on the circle. The circle, there's pis involved, how do you get something without pi talking about circle, kind of doesn't work. So one lap around the circle, if you just pick a nice circle of radius 1, is 2 pi. So you set these two things equal to each other, okay, and then from there, you can get the other angles just by breaking it up. So in particular, let's look at some common angles here. If I have an angle that's at the x-axis, so no angle, so 0 degrees would be 0 radians, that doesn't change. What happens if I go half a lap, so maybe we'll start on the right, and we'll go counterclockwise all the way to the left. So that angle, this is 180 degrees, angle of a straight line. But it's also, running out of room here, it's also half the circle, so half of 2 pi is just pi radians. Now I'm trying to be good here and write my degree symbol and write my radians. A lot of times, you don't see that if you open a random book of math or something like that and you see an angle measurement, it's weird to have degrees in an odd way. So if you see a number, you should assume it's radians unless you see the degree symbol. So I write R-A-D for rad, some books put an R, some people leave it off, but that's for radians. The other one, let's do a quarter turn, so a quarter rotation puts you at the positive y-axis. In terms of degrees, this is 90 degrees, this is the right angle that you know and love. Sometimes it's denoted with a little half square there. So what is this in radians, get used to thinking in radians, well, it's half of pi, so pi over 2, okay? You can also grab the bigger ones, so let's go another angle that goes to the negative y-axis, again, always starting at the positive x-axis, so it's a three-quarter turn. In terms of degrees, what would this be, this would be 180 plus another 90, so that's 270. Again, the goal here is to start thinking about what would this be in terms of radians. Eventually, as you take more math classes, you're going to speak only in radians, so get used to what this is. This is pi, and then another half a pi, so it's like pi and a half, so how do you say 1 and a half? 3 over 2, and then pi, so 3 over 2 pi, okay? So these are your common angles, 0, 180, 90, and 270 are now replaced with 0 radians, pi radians, pi over 2 radians, and 3 pi over 2 radians. Obviously, if we get a more sort of angle handed to us in degrees that is not one that we know just from looking the graph, we'd like some formula that would convert between degrees and radians. So our formula to go from degrees to radians, it's a pretty easy formula. We take our degrees, whatever it is handed to us, and this reminds me of science a little bit. We have the factor label method, anyone remember that, where you want to cancel the units? So you have to cancel degrees, so you want something in degrees down here times radians upstairs. And the idea is, the degrees will cancel and you get only radians in the numerator. And this is called a conversion factor when you make this term. So we pick 180 degrees, and then 180 degrees is pi radians. You have to realize, pi radians, 180 degrees, this is just equal to 1. This is just equal to 1, so this is your conversion factor. Honestly, you're just canceling units. In science, sometimes you do this when you convert from one unit to another. You multiply by something so that the units on the bottom, the degrees sort of cancel, and you're left with radians. So that gets you your radians, so whatever x is, you multiply it by pi over 180. The idea is, how do you remember it's pi over 180? Because I want to land in radicals, I want the units for radicals in my numerator. So let's do an example with something that we just saw, so what's an easy one we can do? How about 360 degrees, so 360 degrees, the full lap. If this thing works out right, I should get 2 pi radians. So we have times our conversion factor of pi radians over 180 degrees. Freeze, this is just a fancy way to write 1. So when we get 360 and 180, well, that was cancel to get 2, you left with a pi and my units is radiant, so 2Ï€ radians as I expected, and this is kind of nice. What's beautiful about this formula is that it works for just some any old choice of degree want then trying to convert it to radians. So let's do one more, how about 135 degrees? Well, that'd be pi radians over 180, and you can cancel and clean this up so there's you get 3 4 times pi, don't lose the pi from the numerator. And I guess if you first started off, its a good thing to just keep reminding yourself why you're doing all this and put your units there. So given degrees, you want to have a way to convert to radians. And sometimes if I give you a picture, as another example, I want you to know looking at something what the radian measure is. So for example, I show the x by axis, I'm going to draw, so imagine something rotating or something that. I'm going to draw an angle that does a full rotation and then I'll hack, it goes again so we can building some sort of spiral here. And then just for fun, let's go all the way to the negative x-axis. So to get this kind of spiral picture here, and I want to know what is the measure of the angle? What is theta? What's the angle measurement? How many rotations have we gone? So let's follow the path here a little bit, so I do one full rotation. So given the picture, find the measurement of the angle, but in radians. So I did one full rotation, so right away full rotation around the circle is 2 pi. I follow the path, I did another full rotation, so there's another 2 pi, so we've got two rotations. And then I've gone half a rotation, so that's pi, so you put it all together 2 pi + 2 pi + pi is good old 5 pi's. So start thinking in terms of radians, this is the hardest part for people. A lot of people, when they first do this, the constantly go back to degrees, is what they're familiar with. But try to get yourself used to talking and thinking of radians. I thought we could make a table, here are some standard angles that we know. We can make a reference, you can use it as a reference sheet. Think of it as your dictionary to translate between degrees in radians. There are obviously lots and lots of angles here, so we can't list all of them, they're infinitely many. But we certainly can list some of the standard ones or the most common ones that you'll see. So for example, it's pretty common to see 0 degrees, maybe little degree symbol. 30 degrees is pretty common, 45 degrees is also pretty common, 60, 90, and then 180. Radians, now again, I want you to check this with the formula if you want, but we got some of these already, so 0 degrees is 0 radians. It's like there's no rotations, that doesn't change. 30 would be pi over 6, 45 is pi over 4, it's a quarter turn, 60 is pi over 3, one-third of 180, so pi over 3, 90 of course, we already saw, it's pi over 2, that's your right angle, and 180 is pi. For symmetry and reflections, you can use this if you have a good foundational knowledge of these working angles then you can find the other angles if they ever go further or more round. But these are mostly your angles here in quadrant one in the XY plane, or both X and Y are positive. And these will come up often, so it's good to have these memorized at some point or just be able to convert fluently between them. And the other part of this to to become fluent in radians is to think in terms of ratings and speak in terms of radiance. So say something now, like what does a try right triangle? Most people say, it's a triangle with a right angle. Okay, well, what does that mean? All right angle has 90 degrees, switch it in like say it out loud, a right triangle is a triangle with an angle measure of pi over 2. Feels a little weird, probably at first, but it's completely accurate and now you're speaking in radiance, so this is the idea. What's an acute angle? Acute angle is something as an angle whose measure is less than pi over 2, so talk in terms of radians, it will help these things sink in, okay? I still soon example, a right triangle has one angle of pi over 3 radians, find other angles. We're going to study triangles a lot, they're very, very useful. They also have lots of nice symmetry and given little bit of information, you can quote unquote solve the triangle. It's always helpful to draw a picture whenever you want these word problems, so get used to drawn triangles here and labeling some things. So we have a right triangle, that means we have a right angle was just said before it's five or two. We have an angle one angle measure of pi over 3. It doesn't matter where you put this thing. I'm going to put at the top because I want to, but you can certainly put every else, find the other angle. Well, there's only one other angle, and now instead of X for the unknown with angles is pretty common to write theta. So it's pretty cool to write theta, it can feel very Greek as you do it. And so now, of course, what's the common fact about angles? Three angles in a triangle, add 2, be careful, be careful, be careful. We're in radians, so before I say it like pause video, save sentence in radians. The sum of the angles of a right triangle of any triangle really, add 2 pi radians. You defaulted there at 180 degrees it's understandable, but try to switch it up, think about pi. So we have pi over 2 + pi over 3 + theta = pi. This is the equivalent statement for angles add up to 180, but now we're in radiant, so it's all pi. This is a good exercise in algebra, I guess something for fractions, we could do both time method on these fractions on the left. If you have seen both time method, it's amazing. You can do the top left times bottom right, so it's 3 pi + the top right times the bottom left is 2 pi all over the common denominator, which is where the boat I gets the same from all over 6. So you got 5 pi over 6, it's nice way to add fractions without doing common denominator, I use it all the time. So we get 5 pi over 6 + theta is pi, brilliant. And then theta = pi- 5 pi over 6, I just move 5, 5 or 6 other side. We could do a common denominator here if you want, so we'll make it 6 over 6. When we add or subtract fractions, don't forget you need common denominator. And then 6 pi- 5 pi is just 1 pi which is pi over 6. So the third angle here is pi over 6. And you should absolutely whenever you get something like this, you can check it, well, how do you check? Well, make sure your answer works. Is it true that pi over 6 + pi over 3 + pi over 2 = pi? Do the three angles add to 180? You can absolutely check these things in here. You can do a common denominator of 6, but it certainly will work. So get in the habit of checking your work, especially if you knew these things or adding subtracting fractions, it's been awhile. Go through the process, check that and you'll see that it works. All right, so great job on this beginning one, keep this table handy between degrees in radians. Start to think and speak in terms of radians, and we'll see you next time.