Final answer:
To find the exact value of cos(7π/12), we use the sum formula for cosine with the values of π/4 and π/6 to get the result (√6 - √2)/4.
Step-by-step explanation:
The expression cos(7π/12) can be evaluated using sum or difference formulas for cosine, as 7π/12 is not a standard angle on the unit circle. Since 7π/12 can be rewritten as (π/4 + π/6), we can use the sum formula for cosine: cos(A + B) = cos(A)cos(B) - sin(A)sin(B). Therefore:
- Identify A and B where A = π/4 and B = π/6.
- Evaluate cos(π/4) = √2/2 and sin(π/4) = √2/2.
- Evaluate cos(π/6) = √3/2 and sin(π/6) = 1/2.
- Substitute these values into the sum formula to get cos(7π/12) = (√2/2)×(√3/2) - (√2/2)×(1/2).
- Simplify the expression to get the exact value.
After simplifying, we find:
cos(7π/12) = (√6/4) - (√2/4) = (√6 - √2)/4.