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

1 Answer
Nghi N
Dec 31, 2016

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

Explanation:

P = sin (pi/8).cos (pi/4)
Trig table gives cos (pi/4) = sqrt2/2, then P can be expressed as:
P = (sqrt2/2)sin ((pi)/8).
We can evaluate sin (pi/8) by applying the trig identity:
2sin^2 a - 1 - cos 2a
2sin^2 (pi/8) = 1 - cos (pi/4) = 1 - sqrt2/2 = (2 - sqrt2)/2
sin^2 (pi/8) = (2 - sqrt2)/4
sin (pi/8) = +- (sqrt(2 - sqrt2)/2)
Since sin (pi/8) is positive, take the positive value.
Finally:
P = ((sqrt2)/2)(sqrt(2 - sqrt2)/2)

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