>> Cosine of 4 pi over 3. Well, here it is, now I'm going to think through it as you might think through it and I'm forgetting a lot of information. Okay. So, I might think, well I know that when pi is involved, I'm talking about radian measure so I might think of this as cosine 4/3rd's pi. And I want to kind of separate my thinking like that because I know that in the coordinate plane, I have some points of pi and 2 pi and those are really important points on the overall coordinate plane when I turn angles of those sizes. Okay, so this then, now, 4/3rd's, 3 goes into 4 one time with 1 left over, so this is 1 and 1/3rd pi. Oh good, now I can think about the coordinate plane and I know that this is pi, this is 0 and it's also 2 pi if I go all the way around. Okay. In terms of radian measure, this is pi over 2. Okay, good. Now 1 and 1/3rd pi, 1 pi would be an angle to this point. 1/3rd beyond that, 1 and 1/3rd pi you see, would take us around to, oh, maybe right in here. Okay, good. Now we have an angle in the 3rd quadrant. Okay. And I have a reference angle right in here and I can think about a reference triangle if I want to. And now, this reference angle, how big is that angle? Well, let's see, this is 1 pi, the whole angle is 1 and 1/3rd pi, so I take 1 and 1/3rd and subtract 1 pi and I'm left with 1/3rd pi. 1/3rd pi. Okay. So our reference angle here is 1/3rd pi or simply pi over 3. Wait a second, I recognize that as one of the familiar angles I was supposed to remember the value of, but I don't remember the value of it. Hmm, what can I do? Well, you know, I kind of have a feeling for the angles in degree measure a little bit better than the angles in radian measure because maybe we've been using those a little more frequently. So, what I can do about that is this, I can think, well let's see, I know that pi corresponds with 180 degrees and even if I don't remember the formula for changing from one kind of angle to another, I can understand the problem this way. Pi over 3 is what I'm trying to figure out. Now, from pi to pi over 3, I'm dividing by 3. If I divide by 3 over here, I find this to be 60 degrees. Oh! This is the 60-degree angle that corresponds with pi over 3. Now, I could put that back into my diagram but I know that cosine of the angle pi over 3 is the same as the cosine of 60 degrees. All right, now, I've got a little situation going on with both the calculation for the cosine of 60 degrees as a reference angle of the triangle and also the sine associated with the situation. Hmm, the sine. Well, let's analyze what the sine pattern is, I forgot what it was. Okay, I know that cosine is adjacent over hypotenuse so I've got the adjacent side is an X value and the hypotenuse is always positive when I'm talking about turning angles and reference triangles and so forth into coordinate planes. The X though is negative so cosine is negative here. Oh, now I remember, it's the X idea and it's the cosine of angles in the 3rd quadrant are negative and also negative in the 2nd quadrant but positive in first and 4th. Okay, I remember that now. All right, so I know this much, that the cosine of this angle is going to be negative and now all I have to do is figure out the cosine of 60 degrees. Well I forgot what it was. Okay, so if I forget what it was, then I'll go to my little reference triangle and my reference triangle might look like this. I've got a 60 degree so, angle here, and now I remember there was also this 30-degree angle up here. And now, gee, I forget the values for the sides. I think it was 1, 2, and the square root of something, but I don't know exactly where they belong. Well, let's go back to the idea of our original discussion with this. Our original discussion started out with an equilateral triangle, a triangle in which all the angles are 60 degrees and all the sides are the same length. Okay? Then we bisected this rascal. We said, okay, if this was a 60-degree angle, then after we bisect, that's 30. Oh, here's our 60 right here. Here's our right angle. Now we just make the sides convenient. It's convenient to make this side 1 and therefore, this is 1 and the whole thing is 2 and that's 2. So we have, oh, here we go, now take all of this out in your imagination, we have 1, 2 and what's this? Well, if we need to, we can go to the Pythagorean relationship and figure it out. It turns out to be the square root of 3. We have the 1, 2, square root of 3, triangle that we refer to in making our calculation. We're thinking about, let's see the cosine of 60 degrees, that's adjacent over hypotenuse, so its 1/2. And therefore, the cosine of 4 pi over 3 is negative 1/2.