- So in this case, now, a couple of things you guys need to understand is when we're looking at this, it all really depends on-- now, you guys can see we're going to use the half-angle formulas to evaluate this. But the half-angle formula is, you guys can see, it's plus or minus. So do we need to provide both of these-- do we need to provide the positive and the negative solutions? And the answer to that is no. You only do the positive or negative depending on where the angle is. 

And this is where sometimes it can get a little tricky because is the sine of 3 pi over 8, is that positive or negative? So since we're in terms of eights-- think about this. This is now 8 pi over 8, which would be pi. So obviously, guys, 3 pi over 8 is still in the first quadrant because half of that would be 4 pi over 8. So we're kind of good with that. 

The only reason why I bring that up is let's pretend this was, I don't know, 13 pi over 8. 13 pi over 8-- this would be 12 pi over 8. 13 pi over 8 would be down here. And therefore, that angle would be negative, correct? Or the sine would be negative, so therefore you'd have to use the negative of the square root with that. But in this case, we're a 3 pi over 8, so we're going to be good. 

All right, so let's go and take a look at the formula. And when we're looking for the sine of u divided by 2, that is simply going to be plus or minus the square root of 1 minus cosine of u divided by 2. Well, obviously, we've already determined that 3 pi over 8 is in the first quadrant. So therefore, sine is going to be positive. 

Now, the next thing though, it's asking, if this is u over 2, that means we need to figure out what u is. That's what we need to plug in. This is u over 2. That's u. This is u over 2. We need to figure out what u is. So let me know if this makes sense. u over 2 is equal to 3 pi over 8. Does that make sense? And once you guys get this, sometimes you can just do this in your head. But do you guys agree with me those angles are the same? Yes? u over 2, 3 pi over 8. 

Now we just go ahead and simplify and multiply by 2 on both sides. And I get 3 times pi times 2 all over 8. I can reduce that to 1/4. u is equal to 3 pi over 4. So the reason why I want to know what u is that's what I'm going to plug into my formula. 

So now I simply just have 1 minus-- what's the cosine of pi over 3 pi over 4? So think 3 pi over 4. That's going to be a negative square root of 2 over 2 divided by 2. Yes, no? Yes? 

STUDENT: Where did the 1 come from? 

- Oh, wait, 1 where? I mean, that's part of the formula. 

STUDENT: Oh. 

- OK. So now let's go and simplify this up a little bit more. I have 1 plus square root of 2 over 2 divided by 2. And I'm using my idea over here to simplify this. I can multiply by the 2 on the top and bottom to get rid of that fraction. So let's multiply it by 2, inside the radical. It's OK, we're doing this inside the radical. We're not changing any values. 

But when I do that, I get 2 plus the square root of 2 over 4. It looks pretty good. However, remember, square roots of radicals, you can take-- like, if I say the square root of a over b, that's equivalent to the square root of a over the square root of b, right? 

STUDENT: Yes. 

- So I can't really take the square root of anything up here, but could I take the square root of in the denominator? Of course I can. So it could be written like that. Or usually we just bring the 1/2 in front. That's kind of it.