use appropriate compound angle formula to determine exact value of 5pi/12

Answer 1

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).