Question

How do you express `sin(pi/ 4 ) * sin( ( 7 pi) / 8 )` without using products of trigonometric functions?

Answer 1

`(sqrt2)(sqrt(2 - sqrt2)/4)`

Explanation:

`P = sin (pi/4).sin ((7pi)/8)`
Trig table --> sin (pi/8) = sqrt2/2
Trig unit circle -->
`sin ((7pi)/8) = sin (pi/8)`
Evaluate sin (pi/8) by using the thig identity:
`cos 2a = 1 - 2sin^2 a`
`cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)`
`2sin^2 (pi/8) = 1 - sqrt2/2 = (2 - sqrt2)/2`
`sin^2 (pi/8) = (2 - sqrt2)/4`
`sin (pi/8) = sqrt(2 - sqrt2)/2` --> `sin (pi/8)` is positive.
Finally:
`P = (sqrt2)((sqrt(2 - sqrt2)]/4)`