Final answer:
To find cos(x/2) given cos(x) = 1/4 and 0 < x < π/2, the positive half-angle formula for cosine is used because x/2 is in the first quadrant. The exact value is √(5/8), with an approximate value of 0.7906.
Step-by-step explanation:
To find cos(x/2), given that cos(x) = 1/4 and x terminates in 0 < x < π/2, we use the half-angle formula for cosine:
- cos(θ/2) = ±√[(1 + cos(θ))/2]
Since 0 < x < π/2, we know that x/2 will also terminate in the first quadrant where cosine values are positive. Hence we take the positive root:
cos(x/2) = √[(1 + cos(x))/2] = √[(1 + 1/4)/2] = √[(5/4)/2] = √[(5/8)]
The exact value of cos(x/2) is therefore √(5/8) or approximately 0.7906.