How do you express sin(pi/ 4 ) * sin( ( 5 pi) / 12 ) without using products of trigonometric functions?

1 Answer
Nghi N.
Jun 24, 2016

(sqrt2/4)sqrt(2 + sqrt3)

Explanation:

Product P = sin (pi/4).sin ((5pi)/12)
Trig table -->
sin (pi/4) = sqrt2/2
P can be expressed as:
(sqrt2/2).sin ((5pi)/12).
We can evaluate sin ((5pi)/12) by using the trig identity:
cos 2a = 1 - 2sin^2 a
cos ((10pi)/12) = cos ((5pi)/6) = -sqrt3/2 = 1 - 2sin^2 ((5pi)/12)
2sin^2 ((5pi)/12 = 1 + sqrt3/2 = (2 + sqrt3)/2
sin^2 ((5pi)/12) = (2 + sqrt3)/4
sin ((5pi)/12) = sqrt(2 + sqrt3)/2 (note: sin ((5pi)/12) is positive)
Finally,
P = (sqrt2/4)(sqrt(2 + sqrt3))

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