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

1 Answer
Nghi N.
Jan 24, 2016

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

Explanation:

First, find sin (pi/8) and cos ((7pi)/6) separately.
Call sin (pi/8) = sin t
Use the trig identity: cos (2t) = 1 - 2sin^2 t
cos (pi/4) = sqrt2/2 = 1 - sin^2 t
2sin^2 t = 1 - sqrt2 = (2 - sqrt2)/2
sin^2 t = (2 - sqrt2)/4
sin (pi/8) = sin t = (sqrt(2 - sqrt2))/2

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

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

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