How do you express sin(pi/ 8 ) * cos(( ( 2 pi) / 3 ) without using products of trigonometric functions?

1 Answer
Nghi N.
Oct 31, 2016

sqrt(2 - sqrt2)/2

Explanation:

P = sin (pi/8).cos ((2pi)/3)
Trig table --> cos ((2pi)/3) = -1/2, then P can be expressed as:
P = - (1/2)sin (pi/8)
We can evaluate the value of sin (pi)/8 by the trig identity:
2sin^2 (pi/8) = 1 - cos ((2pi)/8) = 1 - cos (pi/4) = 1 - sqrt2/2
sin^2 (pi/8) = (2 - sqrt2)/4
sin (pi/8) = sqrt(2 - sqrt2)/2
Only the positive value is accepted because sin (pi/8) is positive
Finally,
P = - (1/2)sin (pi/8) = - (sqrt(2 - sqrt2))/4

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