Find the exact value of the following expression Sin 23 pi12

Find the exact value of the following expression. Sin 23 pi/12 sin 23 pi/12 = (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solution

Sin (23pi / 12) = sin (2pi - pi/12).

Sin (pi/12) = sin (pi/4 - pi/6).

Sin (a - b) = sinAcosB - sinBcosA.

So,

Sin (23pi/12) = sin (2pi) cos (pi/12) - sin (pi/12) cos (2pi).

Notice that Sin (2pi) = 0 and cos (2pi) = 1.

Sin (23pi/12) = - sin (pi/12).

Sin (pi/12) = sin (pi/4 - pi/6) = sin(pi/4)cos(pi/6) - sin(pi/6)cos(pi/4).

==> ( sqrt(2)/ 2) ( sqrt(3) / 2 ) - ( 1/2) ( sqrt(2)/ 2 )

==> sqrt ( 6 ) / 4- sqrt( 2) / 4

==> ( sqrt(6 ) - sqrt(2 ) / 4

sin( 23pi / 12 ) = -- sin( pi /12) ==> --  ( sqrt(6 ) - sqrt(2 ) / 4

==> [ - sqrt(6) + sqrt(2) ] / 4

==> -0.25882