Answer:
In the third quadrant, the sine of an angle is negative, so the angle itself must be between 180 and 270 degrees. This means that the cosine of the angle, and therefore the cosine of half the angle, is also negative.
To find the exact value of the cosine of half the angle, we can use the double angle formula for cosine, which states that cos(2A) = 1 - 2sin^2(A). Since the sine of half the angle is sin(B/2) = sqrt((1 - cos(B))/2), we can plug this into the double angle formula to get:
cos(B/2) = sqrt(1 - cos(B))/sqrt(2) = sqrt((1 - (-1/3))/2) = sqrt(4/6)/sqrt(2) = sqrt(2/3)/sqrt(2) = sqrt(1/3)
Therefore, the value of cos(B/2) is sqrt(1/3), which is positive because B/2 is between 0 and 90 degrees. The exact value of the cosine of half the angle B in the third quadrant is sqrt(1/3).