Final answer:
Given cos(x) = -1/3 and x in the 3rd quadrant, sin(x/2) and cos(x/2) can be found using the half-angle formulas. The correct answer is, sin(x/2) = √(1 + cos(x))/2, cos(x/2) = -√(1 - cos(x))/2, which is option (d).
Step-by-step explanation:
To determine sin(x/2) and cos(x/2) given that cos(x) = -1/3, and knowing that x is in the 3rd quadrant, we need to use the half angle formulas for sine and cosine:
In the third quadrant, sine is positive and cosine is negative, but since we are looking at x/2, we must determine the quadrant for the half angle. If x is in the 3rd quadrant, then x/2 would be in the 2nd quadrant, where sine is positive and cosine is negative. This leads us to choose the negative root for cosine and the positive root for sine.
So the answers are:
Therefore, option (d) is the correct answer: sin(x/2) = √(1 + cos(x))/2, cos(x/2) = -√(1 - cos(x))/2.