Final answer:
The exact value of cos(5pi/4 - 5pi/6) is calculated using the cosine difference identity and simplified to (sqrt(6) - sqrt(2))/4.
Step-by-step explanation:
To determine the exact value of cos(5pi/4 - 5pi/6) using the formula for the cosine of the difference of two angles, we utilize the identity cos(a - b) = cos(a)cos(b) + sin(a)sin(b). We apply this to our specific values:
cos(5pi/4 - 5pi/6) = cos(5pi/4)cos(5pi/6) + sin(5pi/4)sin(5pi/6).
Using unit circle values, we find:
- cos(5pi/4) = -sqrt(2)/2,
- cos(5pi/6) = -sqrt(3)/2,
- sin(5pi/4) = -sqrt(2)/2,
- sin(5pi/6) = 1/2.
So we have:
cos(5pi/4 - 5pi/6) = (-sqrt(2)/2)(-sqrt(3)/2) + (-sqrt(2)/2)(1/2).
Simplifying gives us:
cos(5pi/4 - 5pi/6) = (sqrt(6)/4) - (sqrt(2)/4) = (sqrt(6) - sqrt(2))/4.
The exact value of cos(5pi/4 - 5pi/6) is (sqrt(6) - sqrt(2))/4.