 Hello, and welcome to this screencast for Math 123, Trigonometry. Today we're going to discuss the procedure for finding a reference arc for a given arc. One of the most important things in mathematics is giving very precise definitions of the terminology that we use. And so we're going to take a first look at the definition of a reference arc. Here's the definition. And again, when you read this type of definition, try to pay attention to the important concepts that are introduced. In particular, this definition starts some notation. So we're going to have the reference arc is going to be denoted by x hat. That's typically the way we read that. But more importantly, this arc is going to be in a positive direction. That means we're going to measure it in a counterclockwise direction. And the arc is going to be formed by the terminal point of our arc x and the nearest x-intercept on the unit circle. So down below here is a picture of a unit circle. There are two x-intercepts on the unit circle. One of them is located there. And the other one is located over here on the left. Now the one on the right has coordinates 1, 0. And the one on the left has coordinates negative 1, 0. And one of the things we're going to have to do with an arc is draw it in and see in which quadrant the terminal point lies. And that's where these, what we sometimes call the quadrant arcs become important. The quadrant arcs, one is here and that has a length or measure if you want of pi over 2. And if we go halfway around the circle, remember we're moving in a positive direction counterclockwise, the length of the arc is pi. And if we go 3 quarters of the way around, the length of the arc is 3 pi over 2. And of course if we go all the way around one revolution, we come back with a length of 2 pi. So please keep that in mind as we look at some examples. So here's our first example. We're going to find the reference arc for the arc x equal to 2 pi over 3. And as was mentioned earlier, the first task really is to try to draw the arc. It does not have to be overly precise, but what you want to make sure is that you locate the terminal point and in particular in which quadrant the terminal point lies. So remember we start our measurement at 0.10 and we move positive in a counterclockwise direction. And again remember at this intercept, we have a length of pi over 2. And if we can go halfway around, that has a length of pi. So what we're looking at is 2 pi over 3. So that is an effect looking at if we look at just the fractional part kind of ignoring the pi, we get 1 half is less than 2 thirds is less than 1. So our arc, we're going to have pi over 2 less than 2 pi over 3, which is less than pi. So our arc is going to have a terminal point in the second quadrant. So now we try to draw that in it, try to draw it reasonably accurately. We might stop about there. There's our arc 2 pi over 3. And now what we want to do is determine the reference arc. And remember the reference arc now is going to be measured from the terminal point of our arc to the nearest x-intercept. So in this case, the terminal point is right here marked and the nearest x-intercept is right there. And so our reference arc goes like that and we put the little arrow on it to indicate positive direction. So the blue mark there is our reference arc. And now we have to determine the value of that. And it basically goes back to this idea of using the final point, which is this final value, minus the initial point for the arc, which is at 2 pi over 3. So in calculating our reference arc, this is what we get. X hat equals the final value pi minus the initial value 2 pi over 3. And now we basically try to simplify that. And maybe the easiest way to do that is to factor out the pi and look at the subtraction of 1 minus 2 thirds. And we can see that that's equal to 1 third, so the reference arc is 1 third pi. And as we said, that will be the reference arc for the arc x equal to 2 pi over 3. One of the reasons we're going to be interested in this is trying to find the trigonometric function values, namely sine and cosine, for 2 pi over 3. This reference arc will help us a lot in doing that, but that's coming up in the future. So the thing we want to do, we want to find the reference arc for the arc x equal to negative 3 pi over 4. And now remember, this one is going to move from the point 1, 0 in a clockwise direction, a length of 3 pi over 4. So it gets the negative 3 pi over 4. And although it's possible to work with that value to find the reference arc, it's often easier to use what is called the least positive coterminal arc and use that to determine the reference arc. And it's pretty much what it says. It's a coterminal arc, so in other words, the terminal points will be the same as the one for x. And it's the least positive one. And the least positive one is going from the point 1, 0 clockwise up to 2 pi. So the least positive coterminal arc will satisfy that inequality. So what I'd like you to do now is pause the screencast for a little bit and see if you can determine the least positive coterminal arc for minus 3 pi over 4 and then use that to find the reference arc for x equal to minus 3 pi over 4. Okay, welcome back. Here we'll go through the procedure and hopefully you are able to be successful with this. So as I said, the first task here is to draw the arc. And again, now we're moving in the counterclockwise direction. So at this quadrant, it's minus pi over 2, and at this quadrant, it's minus pi. So our value of 3 4 or minus 3 4 is going to be halfway between minus 1 half and minus 1. So our terminal point for minus 3 pi over 4 will be about there. So here's our arc. And now we want to find the least positive coterminal arc. And that actually is pretty well shown on this picture of the unit circle already. It's got to be positive, so you have to move in a counterclockwise direction. And it has to have a terminal point, or a terminal point, same thing as our arc. And so we start here, and the least positive coterminal arc looks something like that. And now the question is, what's the value of that? One way of doing this is just to start with our value in this case of minus 3 pi over 4 and add one complete revolution to it. In other words, add 2 pi. If necessary, we add more than one complete revolution. But doing that computation, we would see we take minus 3 pi over 4 and add 2 pi. And again, it might be convenient to factor out the pi and work with the fractions. So we write that. And now we have to do the minus 3 force plus 2. And I'll do this pretty quickly. 2 is the same thing as 8 force. So we have 8 force plus a negative 3 force, and we get 5 pi over 4. So that's the least positive coterminal arc. So instead of working with this, we will now work with 5 pi over 4. And now we want to find the reference arc for that. And the reference arc for that now is going to, again, remember be measured in the positive direction to the nearest x-intercept. So it'll actually this time have to start at the x-intercept and move like this. And that would be the least positive coterminal arc. I'm sorry, the reference arc for 5 pi over 4 and the reference arc for minus 3 pi over 4. And using the 5 pi over 4 now, we basically do the same thing. The final value minus the initial value. But now remember, the initial value is going to be not minus pi, but pi, because we're moving in the positive direction. So our x-hat, the reference arc, will be the final value 5 pi over 4 minus the initial value of pi. And again, that's 5 force minus 1 times pi. So x-hat is 1 fourth pi, or if you prefer, you can write it as pi over 4. So that gives you two examples of finding a reference arc. And the second one, we used the concept of the least positive coterminal arc. And that's it for now. See you later.