exact value of sin(-5pi/12)?

1 Answer
Apr 30, 2018

#sin (-{5pi}/12) = -sin (pi/4 + pi/6) #

#= -( sin (pi/4)sin (pi/6) + cos (pi/4) cos (pi/6) )#

# = - ( (sqrt{2}/2) (1 /2) + (sqrt{2}/2) (sqrt{3}/2) )#

#= - 1/4 ( sqrt{2} + sqrt{6}) #

Explanation:

# - {5 pi}/12 = -75 ^circ = - (45^circ + 30^circ)#

This one has both cliches of trig, 30/60/90 and 45/45/90, in one problem.

Plan: Let's start by expressing this using a trig function of something in the first quadrant, and then use the sum angle formula.

#sin (-{5pi}/12) = - sin ({5pi}/12)#

#= -sin (pi/4 + pi/6) #

#= -( sin (pi/4)sin (pi/6) + cos (pi/4) cos (pi/6) )#

# = - ( (sqrt{2}/2) (1 /2) + (sqrt{2}/2) (sqrt{3}/2) )#

#= - 1/4 ( sqrt{2} + sqrt{6}) #