find the exact vavlue of sin 31pi/12 (using sin(alpha-beta)or sin(alpha+beta)?

Answer 1

Well, first get it in the range [0, 2π):

sin(31π/12) = sin(31π/12 - 2π) = sin(7π/12)

Now, 7π/12 = π/4 + π/3, so

sin(31π/12)= sin(7π/12) = sin(π/4 + π/3)
= sin(π/4) cos(π/3) + cos(π/4) sin(π/3)
= [sqrt(2)/2][1/2] + [sqrt(2)/2][sqrt(3)/2] = (sqrt(2) + sqrt(6))/4