Let's start by converting to degrees, which are commonly easier to work with. Use the conversion factor #180/pi#.
#(23pi)/12 xx 180/pi = (23 xx 180)/12 = 345˚#
Now, #345˚# can be written as #300˚ + 45˚#, or for the sake of the problem, #sin(300˚ + 45˚)#.
We must next expand this using the sum formula #sin(A + B) = sinAcosB + cosAsinB#
Write #sin(300˚ + 45˚)# in this form and evaluate.
#sin(345˚)#
#=> sin(300˚ + 45˚) #
#=> sin300˚cos45˚ + cos300˚sin45˚#
#=>-sqrt(3)/2 xx 1/sqrt(2) + 1/2 xx 1/sqrt(2)#
#=>-sqrt(3)/(2sqrt(2)) + 1/(2sqrt(2))#
#=> (1 - sqrt(3))/(2sqrt(2))#
Make sure to rationalize the denominator:
#=> (1 - sqrt(3))/(2sqrt(2)) xx (2sqrt(2))/(2sqrt(2))#
#=>(2sqrt(2) - 2sqrt(6))/(4(2))#
#=>(2sqrt(2) - 2sqrt(6))/8#
#=>(sqrt(2) - sqrt(6))/4#
Hopefully this helps!