find the exact value of the expression 3sin15cos195 2 sin10

find the exact value of the expression. (3sin15)(cos195) + 2 sin105

(3sin15)(cos195) + 2 sin105

(3sin15)(cos195) = 3sin15cos(180+15)

= -3sin15cos15

= -(3/2)sin(2*15) [ use the formula : sin2a = 2sina*cosa]

= (3/2)sin30

= -(3/2)(1/2) = - 3/4

Now, 2sin105 = 2sin(90+15)

=2cos15

use the formula : cos2x = 2cos^2x -1

So, cos15 = sqrt[( 1+ cos30)/2]

= sqrt[(1 +sqrt3/2)*2]

=sqrt(2 +sqrt3)/2

So, 2cos15 = sqrt(2+sqrt3)

So, (3sin15)(cos195) + 2 sin105 = -3/4 + sqrt(2 +sqrt3)

= [-3 +4sqrt(2 +sqrt3)]/4