Final answer:
To find the value of cos(11π/12), we use the cosine addition formula and the unit circle values for cos(3π/4), sin(3π/4), cos(π/6), and sin(π/6). After simplifying, we get cos(11π/12) = -(√6 + √2)/4, which does not match any of the given options.
Step-by-step explanation:
To find the value of cos(11π/12) using addition and subtraction formulas, we can break down the angle into the sum or difference of angles we know the sine and cosine values for. One way to express 11π/12 is as the sum of π/4 and π/6, since 11π/12 = 3π/4 + π/6.
Using the cosine addition formula:
cos(α ± β) = cos α cos β - sin α sin β
We get:
cos(11π/12) = cos(3π/4 + π/6) = cos(3π/4)cos(π/6) - sin(3π/4)sin(π/6)
The known values from the unit circle are cos(3π/4) = -√2/2, sin(3π/4) = √2/2, cos(π/6) = √3/2, and sin(π/6) = 1/2.
Substitute these values into the equation:
cos(11π/12) = (-√2/2)(√3/2) - (√2/2)(1/2)
Simplify the expression:
cos(11π/12) = -√6/4 - √2/4
cos(11π/12) = -(√6 + √2)/4
Knowing that -(√6 + √2)/4 is approximately -0.9659, we see that none of the options provided in the question is correct. Therefore, the correct value does not match any of the options given (a) through (d).