Find the exact value by using sum identity sin(11pi/12)

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sin 11pi/12 = sin (6pi/12 + 5pi/12) =

=sin(pi/2 + 5pi/12) = sin pi/2 cos 5pi/12 + sin 5pi/12 cos pi/2

you know cos pi/2 = 0 and sin pi/2 = 1

= cos 5pi/12

you know the double angle relation:
cos2x = 2cos²x - 1, therefore:

cos 2(5pi/12) = 2cos² 5pi/12 - 1

2cos²5pi/12 = cos 2(5pi/12) + 1 = cos 5pi/6 + 1 = 1 - √(3)/2

cos 5pi/12 = √[(1 - √(3)/2)/2] and
cos 5pi/12 = -√[(1 - √(3)/2)/2]