 In this video, we present the solution to question number eight for practice exam number two for math 1060, in which case we're told that cosine of some angle a is equal to one third. And we know that a resides between three pi halves and two pi so it's in the fourth quadrant. We need to compute cosine of a halves so the half angle identity is going to come into play here. If a is between three pi halves and two pi, then that means that a halves will reside between two pi divided by two, which is just pi. And three pi halves divided by two, which is three pi force, right? So which quadrant is this in now? Notice that a halves is going to live inside of the second quadrant. That'll be helpful for us because by the half angle identity cosine of a halves, this equals plus or minus the square root of one plus cosine of a over two, where this plus or minus is dependent upon the quadrant that we're in. So since we're in the second quadrant, cosine is going to be negative there. We also know that cosine of a is equal to one third, so we get one plus a third over two. To clean up these fractions, I'm going to times the top and bottom of the big fraction by three, you're going to distribute it like so. This then gives us negative the square root of three plus one over six, for which we then get the square root of four over six. Since four is a perfect square, I can take the square root of the top to get negative two over the square root of six, like so. And then this leads me to choose choice B as the correct response.