 Welcome back. In the last video we talked about strategies for actively reading a textbook. In brief they were take notes especially on definitions. Work out examples fully for yourself and always ask why and answer that question for any statement the textbook makes. Summarize your thoughts and look out for big picture ideas. In this video we'll be putting those ideas to use. As a reminder electronic note-taking is okay, but writing or about typing is not. Writing helps you retain information better. I'm going to be taking notes on page 28 of the math one two three textbook trigonometry. So here that is over here, and here's my note paper on this side. You're of course welcome to use an iPad or any other similar device. So let's get started. We define the reference arc x hat. Okay, well I clearly need to take notes on that. It's a definition. So reference arc and it's called x hat. Although I'm not sure what that means. Alright, so reference arc x hat. It's an arc with positive direction formed by the terminal point of the arc x and the nearest x intercept on the unit circle. Okay, there's a lot of things there. I see some examples coming up. So I'm going to take notes on the key parts of this. It's an arc with positive direction. It's formed by okay, so its end points are the terminal, the end of x, the terminal point of x, and the nearest x intercept. So there is something we're calling x. x is the name of a of a arc and x hat is the name for this new thing we're making that's somehow related to it. That's all that I understand so far. The length of the reference arc x hat is therefore the length of the smallest arc along the circle from the terminal point of x to the x-axis. And I've already put a mark here as you can see because I didn't understand that the first time I read it. I don't understand it now, so I'm going to have to take another look at that. But I see there's examples, so I'm going to try that. Okay, so for example, excellent. Alright, for example, I'm going to write down the example. The arc x equals 3 pi over 2, let's draw that, the arc 3 pi over 4 I mean, terminates in quadrant 2. So 3 pi over 4, I know that it's most of a semi circle, but it's it's a half or it's a quarter pi over 2 and then another quarter. So 3 pi over 4. So it goes out to here. So this is x. It terminates in quadrant 2. Sure, I agree with that. So it's reference arc, the arc from the terminal point of 3 pi over 4 to the x-axis. Okay, its reference arc has positive direction. So I'm going to look at this definition. It has positive direction, so it's got to go that way. Its end points include the terminal point of x, which is right there, and the nearest x intercept. Okay, and I know the unit circle crosses down here. That's as close as I can get because the unit circle crosses here and there and that's it. So I've got to go here to there. This is what the definition is telling me. So this must be the reference arc we're talking about. Okay, I'm looking at this example over here. I can see I basically recreated the picture. There's x. There's x hat. This is the thing called the reference arc. All right. And so that's this is the reference arc. Okay, so back to what I was reading. It's reference arc, the arc from the terminal point of 3 pi over 4 to the x-axis. So that's what I found. Good. Must be pi minus 3 pi over 4. What does that mean? I know pi is this length. And we were given 3 pi over 4. So I'm taking the whole length minus that. So I'm getting just this part left over. And that I can do the arithmetic is pi over 4. Okay, so they're saying this arc is pi over 4 long. So it's sort of a it's a shorter arc. It's like the rest of the semi circle. So this is a good summary. This looks like x hat is the rest of the semi circle. Okay, well, that's something. The arc x equals 11 pi over 6. Okay, it's another example. So the arc 11 pi over 6. Let's draw that. I know that that is big. That's most of a circle. Because 12 pi over 6 would be the whole circle. So 11 pi over 6 is most of the way around. And I'm just going to sketch as much of it as I need. Its reference arc would be what? So I'm going to actually do this before I read what's being said there. So it's arc with positive direction. So it's got to go this way. Its endpoints include the terminal point of x. Okay, this is x. And the nearest x intercept, which is definitely here. Okay, so it's got to just be this. This is my reference arc. It's a really short little thing. And sure enough, that's what the picture over here says. So this is x hat. It's reference arc. The arc from the terminal point of 11 pi over 6 to the x axis. That's what I drew. It must be 2 pi, that's the whole circle, minus 11 pi over 6. So I'm just going to label this. This is the whole circle. This is x. This is the arc we're given. And I'm taking the whole circle and getting rid of the whole bit here. And so I'm left with just this little piece, the reference arc. Okay, and that's what? That's 12 pi over 6 minus 11 pi over 6, which is pi over 6. So it's saying this thing is, again, it's, okay, this time x hat is the rest of the circle. So we're sort of completing, right, we're completing complete part of the circle. I think that's what I'm getting from it so far. I might be wrong. I can come back and edit this later. Okay. So the following figures show reference arcs for arcs x within one wrap of the circle, zero less than x less than 2 pi. Okay, so that just means we're not going too far around the circle. Okay, that makes sense. So I'm actually going to look at these on this paper, maybe take notes on here right now. This is another good strategy to use the book to take notes. So this one right here. So there's x, I see. There's x hat. And do I agree that that's x hat? It says x hat equals x. Do I really agree with that? I'm actually going to move this over and zoom in a little bit to point out what I'm looking at here. So going back to the definition again, so I'll move this definition down here. Reference arc is the arc with positive direction that way. Endpoints include the terminal point of x. So that one, and the nearest x intercept, which really ought to be over here. Oh, except that's not x. That's y axis. This is the y axis. This is the x axis. We've got a lot of x's floating around. I should be careful about that. And so it's got to be the arc involving the terminal point of x and the nearest x intercept, which is that one. Okay, so yeah, so the reference arc is is the same thing. So something I'm noticing here is this looks like a short arc, right? This is an arc that's, it's just like x, but short and this arc was already short. So I'm going to say this is a short arc is the thing I've noticed about it. Okay, this one looks exactly like the example up here. I've seen this. So what does it say? It says it's the arc in this direction. Endpoints include the terminal point of x. Sure. And the nearest x intercept there. Okay, I've already found x hat. It says the arc to the x axis. Sure. So that's how I get to the x axis. It's the rest of the semicircle. And its length is pi minus x. Okay, I already saw that. It's the whole semicircle minus this bit. So it's the rest of the way to the semicircle. So it's the rest of the way to the x axis here. Okay, that makes sense. Right here. Oh, this is a long one. I haven't seen this before. So this arc goes to here. And I'm getting used to saying it's the arc in the positive direction is going this way. It has to involve the end of x and the nearest x intercept, which is not here, that's too far. So it's got to be here is the closer one. Okay, so saying x hat is that. All right. And again, this is sort of, it's not the rest of the way to the x axis, but it's to the closest x intercept. So I'm sort of getting the part of the semicircle that looks most like the bit that just goes to the x axis. It's the shortest bit to the x axis. That's a good definition. So here the reference arc is the original one minus pi. What's that? Oh, this is pi. So I'm just cutting away the bit that doesn't matter. And leaving only the bit that goes right to the x axis. It's interesting that they overlap that's something I hadn't seen before. So they can overlap. So I should make a note about that can overlap. Okay, the last one, oh, this looks just like the one up here. All right. So we have a long arc, I go the rest of the way to the x axis. And it's the whole circle minus my length. Okay, that makes sense. Let's see. Okay, so I've worked through those examples, I've asked myself why all the time here. So I'm going to rearrange this and zoom back out a little bit. So that we can go back to taking notes says as a result of the definition, the length of any reference arc is always less than pi over two. Which is what this says. Do I believe that? I mean, certainly all these examples I've seen have short reference arcs. Okay, but it says I always have to go to the x axis. There's always going to be a jump to the x axis nearby. Oh, yeah. And if I'm over here, I'm going to jump to this one. If I'm over here, I'm going to jump to that side. If I'm over here, I'm going to jump to there. And here, I'm going to jump to there. So pi over two is just a quarter of a circle, quarter of a wrap. And so I can jump to the nearest x intercept without going further than a quarter of a wrap, I believe that that makes sense. Okay, caution. Ah, caution. Okay, so I had better darn well take notes on that. So the caution is be careful to remember that we find reference arcs with respect to the x axis, not the y axis. Okay, that's a mistake I just made. So it's go to x axis, not y axis. So right, so the picture is saying I go like this, not like this. I don't continue on there. Okay, that makes sense.