Final answer:
Using the half-angle identity for cosine, the positive value of cos(1/2 B) with cos(B) = 2/3 is found to be √[30]/6 in simplest radical form with a rational denominator.
Step-by-step explanation:
If we have cos(B) = 2/3, and we are looking for the positive value of cos(1/2 B), we can use the half-angle identity for cosine, which states:
cos(1/2 B) = ±√[(1 + cos(B))/2]
Substituting the known value of cos(B):
cos(1/2 B) = ±√[(1 + 2/3)/2] = ±√[(3/3 + 2/3)/2] = ±√[5/6]
We take the positive square root because the question asks for the positive value:
cos(1/2 B) = √[5/6] = √[5]/√[6] = √[5]/√[6] * (√[6]/√[6]) = (√[5] * √[6])/6 = √[30]/6
Thus, the positive value of cos(1/2 B) in simplest radical form with a rational denominator is √[30]/6.