>> In this next one, very similar idea. We have cosine X over 2 equals 1 over the square root of 2. And you begin the analysis by kind of thinking like this, that this is cosine one-half X. Let's think about it like that... and if we're saying that X is falling between 0 and 2 pi, then what about one-half X? Well taking one-half and all parts
[ Writing on the Board ] ... then one-half X is falling between 0 and pi. Alright, let's keep that in mind as we go through our analysis. And now we just kind of follow our nose through it as we normally would. The cosine of some angle, and we don't really care how this is expressed at the moment, but the cosine of some angle is 1 over the square root of 2. Now this was the one associated with the 45 degree angle, or the angle of pi over 4. And so we're thinking about... the one-half X then is pi over 4. Now that's in the first quadrant, but where else is cosine positive? Hmmm, cosine. Let's see, cosine is associated with X and that's the 1st quadrant and the 4th quadrant. Okay, 1st quadrant and 4th quadrant, so we have a reference angle of pi over 4 in the 4th quadrant. Now in the past what I have been doing is this, I've been thinking that... we take our
[ Writing on the board ] ... we take the unit circle, and we're thinking cosines, so cosine... sign pattern for cosine is like this. I have a reference angle of pi over 4 in that quadrant, and also this quadrant. Actually it's an actual angle here, it's a reference angle here. And now I'd like the angle associated with this reference angle of pi over 4. So I'm thinking about let's see... I've got pi right here. This is 4 pi over 4, 5 pi over 4, 6 pi over 4, 7 pi over 4. Okay, this is 7 pi over 4 for that angle. Now, so those are my 2 angles that solve this, but remember that one-half X is pi over 4. And one-half X is 7 pi over 4. Now I was considering 1 complete circuit here, but let's kind of look at our reference to our domain for the equation that we're dealing with. Hmmm, this says that 0 is less than one-half X, is less than pi - oh, the one-half X has to fall between 0 and pi. And this value for one-half X doesn't fall in there, so we need to throw that out. And this then becomes our only solution, but we're not finished because we want the value of X, not the value of one-half X. So let's see - associated with this, if I multiply by 2 on both sides I find, let's see, X to be equal to, multiply this by 2 and we have 2 pi over 4, which is simply pi over 2.