The problem gives us the sine values of two angles - x and y - and asks us to calculate the cosine of their difference. We know that sin x = 4/5 and sin y = 1/2.
Firstly, let's find the cosine values for x and y. We can do that by using the Pythagorean identity that correlates sine and cosine, which is sin²x + cos²x = 1.
Now, let's calculate cos x. We know that x lies in quadrant II, and in that quadrant, cosine has a negative value. So, we subtract sin²x from 1 and take the negative square root to get cos x:
cos x = -sqrt(1 - sin²x) = -sqrt(1 - (4/5)²) = -0.6
Then calculate cos y. We know that y lies in quadrant I, and in that quadrant, cosine has a positive value. So, similar to what we did above, we subtract sin²y from 1 and take the positive square root:
cos y = sqrt(1 - sin²y) = sqrt(1 - (1/2)²) = 0.87 (rounded to two decimal places)
After calculating cos x and cos y, we can calculate cos(x-y) using the cosine subtraction identity, which is cos(x-y) = cos x cos y + sin x sin y:
cos(x-y) = cos x cos y + sin x sin y = (-0.6) * (0.87) + (4/5) * (1/2) = -0.12 (rounded to two decimal places)
So, cos(x-y) = -0.12.