Final answer:
To find the exact value of cos(π/12), we use the cosine difference formula, identifying that π/12 equals π/3 minus π/4, and substitute known cosine and sine values for these angles to get the result (√(2) + √(6))/4.
Step-by-step explanation:
The student is asking how to find the exact value of cos(π/12) using the sum and difference of cosines formula. The angle π/12 can be represented as (π/3 - π/4), which are angles that we usually know the sine and cosine values for. By using the cosine difference formula, cos(a - b) = cos(a)cos(b) + sin(a)sin(b), we can evaluate cos(π/12).
Step 1: Recognize that π/12 = π/3 - π/4.
Step 2: Use the cosine difference identity with a = π/3 and b = π/4.
Step 3: Calculate:
cos(π/12) = cos((π/3) - (π/4)) = cos(π/3)cos(π/4) + sin(π/3)sin(π/4).
Step 4: Substitute the known values:
cos(π/3) = 1/2, sin(π/3) = √(3)/2, cos(π/4) = sin(π/4) = √(2)/2.
Step 5: Perform the calculation:
cos(π/12) = (1/2)(√(2)/2) + (√(3)/2)(√(2)/2)
= (√(2) + √(6))/4.
Thus, the exact value of cos(π/12) is (√(2) + √(6))/4.