How do you express sin(pi/ 4 ) * sin( ( 7 pi) / 8 ) sin(π4)sin(7π8) without using products of trigonometric functions?

1 Answer
Apr 7, 2016

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

Explanation:

P = sin (pi/4).sin ((7pi)/8)P=sin(π4).sin(7π8)
Trig table --> sin (pi/8) = sqrt2/2
Trig unit circle -->
sin ((7pi)/8) = sin (pi/8)sin(7π8)=sin(π8)
Evaluate sin (pi/8) by using the thig identity:
cos 2a = 1 - 2sin^2 acos2a=12sin2a
cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)cos(π4)=22=12sin2(π8)
2sin^2 (pi/8) = 1 - sqrt2/2 = (2 - sqrt2)/22sin2(π8)=122=222
sin^2 (pi/8) = (2 - sqrt2)/4sin2(π8)=224
sin (pi/8) = sqrt(2 - sqrt2)/2sin(π8)=222 --> sin (pi/8)sin(π8) is positive.
Finally:
P = (sqrt2)((sqrt(2 - sqrt2)]/4)P=(2)(224)