How do you express #cos(pi/ 4 ) * sin( ( 15 pi) / 8 ) # without using products of trigonometric functions?

1 Answer
Jan 24, 2016

#- (sqrt2/2)(sqrt(2 - sqrt2)/2)#

Explanation:

Find #cos (pi/4#) and #sin ((15pi)/8)# separately
Table for trig functions of Special Arcs gives --> cos (pi/4) = sqrt2/2.

#sin ((15pi/8) = sin (-pi/8 + 2pi) = -sin (pi/8).#
Find sin (pi/8)
Use trig identity: # cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)#
#2sin^2 (pi/8) = 1 - sqrt2/2 = (2 - sqrt2)/2#
#sin (pi/8) = sqrt(2 - sqrt2)/2# (pi/8 is in Quadrant I)
Finally
#cos (pi/4).sin( (15pi)/8) = (sqrt2/2)(-sin (pi/8)) =*#
#= - ((sqrt2)/2)(sqrt(2 - sqrt2)/2)#