Suppose x 12 Use the indicated identity to find the exact va

Suppose x= /12. Use the indicated identity to find the exact value of each of the following

a) sin x, usin the difference identity

b) cos x, using the half- angle identity

Solution

x =pi/12

a) sin(pi/12) =

sin(pi/12) = sin[(pi/3)-(pi/4)] = sin(pi/3)cos(pi/4)-cos(pi/3)sin(pi/4)

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

=( sqrt6 -sqrt2 )/4

b) cos(pi/12)

cos^2x = (cos2x +1)/2

cosx = sqrt[ (1+cos2x)/2]

cos(pi/12) = sqrt[ (1+cospi/6)/2]

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

= sqrt(2+sqrt3)/2