How do you express sin(pi/ 8 ) * cos(( 7 pi)/6 ) sin(π8)cos(7π6) without using products of trigonometric functions?

1 Answer
Jan 24, 2016

-(sqrt3)(sqrt(2 - sqrt2)/4)(3)(224)

Explanation:

First, find sin (pi/8)sin(π8) and cos ((7pi)/6)cos(7π6) separately.
Call sin (pi/8) = sin tsin(π8)=sint
Use the trig identity: cos (2t) = 1 - 2sin^2 tcos(2t)=12sin2t
cos (pi/4) = sqrt2/2 = 1 - sin^2 tcos(π4)=22=1sin2t
2sin^2 t = 1 - sqrt2 = (2 - sqrt2)/22sin2t=12=222
sin^2 t = (2 - sqrt2)/4sin2t=224
sin (pi/8) = sin t = (sqrt(2 - sqrt2))/2sin(π8)=sint=222

cos ((7pi)/6) = cos (pi/6 + pi) = - cos (pi/6) = - sqrt3/2cos(7π6)=cos(π6+π)=cos(π6)=32

Finally, sin (pi/8).cos ((7pi)/6) = - (sqrt3)(sqrt(2 - sqrt2)/4)sin(π8).cos(7π6)=(3)(224)