use appropriate compound angle formula to determine exact value of sin13pi/12
To determine the exact value of sin(13π/12), we can use the compound angle formula for sin(A+B):
sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
In this case, A = π/4 and B = π/3. Substituting these values into the formula:
sin(13π/12) = sin(π/4 + π/3)
Using the formula, we can expand sin(π/4 + π/3) as follows:
sin(π/4 + π/3) = sin(π/4)cos(π/3) + cos(π/4)sin(π/3)
Using the values of sin(π/4) = √2/2 and cos(π/4) = √2/2, as well as sin(π/3) = √3/2 and cos(π/3) = 1/2, we can substitute these values into the equation:
sin(13π/12) = (√2/2)(1/2) + (√2/2)(√3/2)
Simplifying the expression further:
sin(13π/12) = (√2/4) + (√6/4)
Combining the two terms:
sin(13π/12) = (√2 + √6)/4
Therefore, the exact value of sin(13π/12) is (√2 + √6)/4.