Evaluate giving exact values and identities. Calculator based solutions are not acceptable. Thanks for any help
cos (7pi/12)
7pi/12 radians = 105º = (60-45)º
Now use the same method I showed you several posts above this one.
To find the exact value of cos(7π/12), we will use the half-angle formula for cosine.
The half-angle formulas are given as follows:
cos(θ/2) = ± √((1 + cosθ)/2)
sin(θ/2) = ± √((1 - cosθ)/2)
tan(θ/2) = sinθ/(1 + cosθ)
We can use the half-angle formula for cosine twice to evaluate cos(7π/12). First, let's write 7π/12 as the sum of two angles:
7π/12 = (5π/12) + (2π/12)
Now, using the half-angle formula for cosine, we can write:
cos(7π/12) = cos((5π/12) + (2π/12))
cos(A + B) = cosA cosB - sinA sinB
cos(7π/12) = cos(5π/12) cos(2π/12) - sin(5π/12) sin(2π/12)
cos(7π/12) = cos(5π/12) cos(π/6) - sin(5π/12) sin(π/6)
Now, we can use the half-angle formulas for sine and cosine to find the values:
cos(π/6) = √((1 + cos(π/3))/2)
cos(π/6) = √((1 + 1/2)/2)
cos(π/6) = √(3/4)
cos(π/6) = √3/2
sin(π/6) = √((1 - cos(π/3))/2)
sin(π/6) = √((1 - 1/2)/2)
sin(π/6) = √(1/4)
sin(π/6) = 1/2
Now, let's find the values of cos(5π/12) and sin(5π/12) using the half-angle formulas again:
cos(5π/12) = √((1 + cos(π/6))/2)
cos(5π/12) = √((1 + √3/2)/2)
cos(5π/12) = √((2 + √3)/4)
cos(5π/12) = √((2 + √3))/2
sin(5π/12) = √((1 - cos(π/6))/2)
sin(5π/12) = √((1 - √3/2)/2)
sin(5π/12) = √((2 - √3)/4)
sin(5π/12) = √((2 - √3))/2
Finally, we can substitute these values into our initial equation:
cos(7π/12) = (cos(5π/12) cos(π/6)) - (sin(5π/12) sin(π/6))
cos(7π/12) = (√((2 + √3))/2)(√3/2) - (√((2 - √3))/2)(1/2)
Now we can simplify this expression by multiplying and simplifying the terms:
cos(7π/12) = (3√(2 + √3))/4 - (√(2 - √3))/4
Therefore, the exact value of cos(7π/12) is [(3√(2 + √3))/4 - (√(2 - √3))/4].