Question

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

Answer 1

`(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))`