Question #42131

1 Answer
Feb 10, 2017

#sqrt(2 + sqrt3)/2#

Explanation:

Unit circle and property of complement arcs -->
#sin ((5pi)/12) = sin (-pi/12 + (6pi)/12) = #
#= sin (-pi/12 + pi/2) = cos (pi/12) #
Find #cos (pi/12)# by using trig identity:
#2cos^2 a = 1 + cos 2a#
In this case:
#2cos^2 (pi/12) = 1 + cos (pi/6) = 1 + sqrt3/2 = (2 + sqrt3)/2#
#cos^2 (pi/12) = (2 + sqrt3)/4#
#cos (pi/12) = +- sqrt(2 + sqrt3)/2#
Since #pi/12# is in Quadrant 1, take the positive value.
Finally,
#sin ((5pi)/12) = cos (pi/12) = sqrt(2 + sqrt3)/2#