How do you express sin(pi/ 6 ) * cos( (3 pi) / 8 ) without using products of trigonometric functions?

1 Answer
Nghi N.
Mar 17, 2016

sqrt(2 - sqrt2)/4

Explanation:

P = sin (pi/6).cos ((3pi)/8).
Trig table --> sin pi/6 = 1/2
Find cos ((3pi)/8) by the identity: cos 2a = 2cos^2 a - 1
cos ((6pi)/8) = 2cos^2 ((3pi)/8) - 1.
cos ((3pi)/4) = -sqrt2/2 = 2cos^2 ((3pi)/8) - 1
2cos^2 ((3pi)/8) = 1 - sqrt2/2 = (2 - sqrt2)/2
cos^2 ((3pi)/8) = (2 - sqrt2)/4
cos ((3pi)/8) = sqrt(2 - sqrt2)/2 (since cos ((3pi)/8) is positive.
Finally,

P = (1/2)(sqrt(2 - sqrt2)/2) = (sqrt(2 - sqrt2)/4)

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