use appropriate compound angle formula to determine exact value of 5pi/12
To determine the exact value of 5π/12, we can use the compound angle formula for cosine, which states:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Let's rewrite 5π/12 as the sum of two angles:
5π/12 = π/4 + π/6
Now, we can use the compound angle formula with A = π/4 and B = π/6:
cos(5π/12) = cos(π/4 + π/6)
Using the compound angle formula:
cos(5π/12) = cos(π/4)cos(π/6) - sin(π/4)sin(π/6)
We know the exact values of cos(π/4) and cos(π/6) from the unit circle:
cos(π/4) = 1/√2
cos(π/6) = √3/2
We also know the exact values of sin(π/4) and sin(π/6) from the unit circle:
sin(π/4) = 1/√2
sin(π/6) = 1/2
Substituting these values:
cos(5π/12) = (1/√2)(√3/2) - (1/√2)(1/2)
Simplifying:
cos(5π/12) = √3/2√2 - 1/2√2
Rationalizing the denominator:
cos(5π/12) = (√3 - 1) / (2√2)
Therefore, the exact value of 5π/12 is (√3 - 1) / (2√2).