 So, we'll talk about the one true method of measuring angles, and by true method I mean actually useful for things like calculus. So, what do we mean by the measure of an angle? Well, the measure of an angle emerges as follows. Imagine that I'm sitting at the vertex of an angle, so here I have an angle, and I'm going to place my observer at the vertex, and I'm going to have them facing along one leg of the angle, so they are facing, for example, this way. The measure of the angle is the amount of rotation they need to undergo in order for them to face the other leg. So, what that means is they're going to turn and rotate around themselves until they face the other leg. So, they're going to turn, turn, turn, and there they are. And so, the angle measure is going to be whatever that amount of rotation is going to be. And what this means is two things. First of all, the natural unit of measures for angles is the turn, and in general we're going to let a counterclockwise rotation correspond to a positive angular measure. So, for example, let's measure, let's draw an angle with a measurement of one quarter of a turn. And in order to draw this, it'll be helpful to have some benchmark rotations, which is to say some standard rotations that I can use as reference points. So, for example, a good one is that imagine I'm at the origin, I'm facing along the positive x-axis, so here I am at the origin facing along the positive x-axis, and let's say I do a full turn. So, I'm now going to do a full turn counterclockwise, here we go. And at the end of that, the thing to notice is that I am back facing the same direction I was before. So, now I'm going to try to do a half turn, and I'll do my half turn, and I end up facing along the negative x-axis. So, now we're ready to do our quarter turn. Our full turn took us all the way around facing the same way. Our half turn took us halfway around facing the opposite direction. And now I'm going to do a quarter turn, so that's going to take me from here, quarter turn counterclockwise, takes me to here, and there's my angle, is going to look something like that. Now, the turn is actually too big a unit of measure to be convenient, we'll always be dealing with fractions, so a much more natural way to measure angles is to use radians. Now, you have probably been introduced to this measure of angle called a degree, and by far the least natural way to measure angles, and the absolute worst possible way to measure angles is to use degrees. We don't ever want to use degrees outside of, well, really the time that you got introduced to using degrees. In everything else, you really want to use radians, and much of the reason for using radians is that if you use radians, the calculus of trigonometric functions is far, far, far, far easier than if you use degrees. So, radian measures work as follows. Suppose I have a circle with a radius equal to 1, so here's my circle with radius 1, I need a name for a circle with a radius of 1, so we'll call this the unit circle, and what I'm going to do is I'm going to walk along the circle some distance theta, and so for example, I might take a walk along the circle, maybe it'll look something like that, and the idea here is that the person who's watching me from the center of the circle will have seen me, and if they're tracking me, what will have happened is they will have turned through some angle, and that angle is going to be this angle theta, and so as I walk along the circle, the person who's watching me undergoes a rotation through an angle of theta. So, for example, let's consider the problem, we want to draw an angle with the measure of pi over 3 radians, as before it's useful to have a few benchmarks to get a handle on what these angles look like, so the first thing we notice is that if I walk all the way around the circle, so there's my walk, I end up there, if I walk all the way around the circle, I'll have gone a distance equal to the circumference of the circle, and that's going to be a distance of 2 pi radians. Well, I don't want to go 2 pi radians, so let's consider another possibility, so maybe this time I'll only walk halfway around the circle, so I'll take my walk around the circle, halfway around the circle, and if I do that, well, a full rotation is 2 pi radians, a halfway around the circle is going to be pi radians. Well, I don't actually want to travel pi radians, I want to travel pi over 3 radians, so what this suggests is that I only want to walk one-third of halfway around the circle. So here I've gone a little bit too far, this is halfway around the circle, I really want to go one-third of that distance, so let's back up a little bit, back up, back up, and there we are. So we back up to about that point, and there's our angle with a measure of pi over 3 radians. Well, let's try one more, we'll sketch an angle with a measure of 5 pi over 6. So the natural unit of angular measure is the turn, so it's probably easiest to think about what fraction of a turn we get for 5 pi over 6. So remember a full turn, once around the circle is equal to the circumference of a circle of radius 1, that's 2 pi. So I can start by writing down one turn is equal to 2 pi radians, and to get pi by itself, I'll divide both sides by 2, that means half a turn is pi radians, and I actually want 5 pi over 6 radians, so I'll multiply both sides by 5 pi over 6, and that tells me that 5 pi over 6 is going to be 5 6 of half a turn. So let's see how I might draw that out. So what I might do is I might take my, so here's my unit circle, here's my pointer, and I'm going to take a walk along the circle, and it's easiest for human beings if we think about this as going to a certain place. If I go halfway around the circle, that's going to place me there, I don't want to go that far, I don't want to do a half turn, I want to do 5 6 of a half turn. So let's go not quite halfway around, not quite halfway around, but 5 6 of halfway around, and that looks something about like that. And so there's my angle that I'm going to form, which is this angle right here in the center.