use appropriate compound angle formula to determine exact value of cos17pi/12
To determine the exact value of cos(17π/12), we can use the compound angle formula for cos(A + B):
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
In this case, let A = 3π/4 and B = π/6, so we can rewrite the angle 17π/12 as 3π/4 + π/6:
cos(17π/12) = cos(3π/4 + π/6)
Using the compound angle formula, we have:
cos(17π/12) = cos(3π/4)cos(π/6) - sin(3π/4)sin(π/6)
Now, let's find the values of cos(3π/4), cos(π/6), sin(3π/4), and sin(π/6) individually:
cos(3π/4) = -√2/2
cos(π/6) = √3/2
sin(3π/4) = √2/2
sin(π/6) = 1/2
Substituting these values into the formula, we get:
cos(17π/12) = (-√2/2)(√3/2) - (√2/2)(1/2)
cos(17π/12) = -√6/4 - √2/4
Simplifying further, we obtain:
cos(17π/12) = (-√6 - √2)/4
Therefore, the exact value of cos(17π/12) is (-√6 - √2)/4.