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

1 Answer
Nghi N.
Apr 7, 2016

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

Explanation:

P = sin (pi/4).sin ((7pi)/8)
Trig table --> sin (pi/8) = sqrt2/2
Trig unit circle -->
sin ((7pi)/8) = sin (pi/8)
Evaluate sin (pi/8) by using the thig identity:
cos 2a = 1 - 2sin^2 a
cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)
2sin^2 (pi/8) = 1 - sqrt2/2 = (2 - sqrt2)/2
sin^2 (pi/8) = (2 - sqrt2)/4
sin (pi/8) = sqrt(2 - sqrt2)/2 --> sin (pi/8) is positive.
Finally:
P = (sqrt2)((sqrt(2 - sqrt2)]/4)

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