Question

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

Answer 1

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

Explanation:

`P = sin (pi/8).cos (pi/4)`
Trig table gives `cos (pi/4) = sqrt2/2`, then P can be expressed as:
`P = (sqrt2/2)sin ((pi)/8)`.
We can evaluate `sin (pi/8)` by applying the trig identity:
`2sin^2 a - 1 - cos 2a`
`2sin^2 (pi/8) = 1 - cos (pi/4) = 1 - sqrt2/2 = (2 - sqrt2)/2`
`sin^2 (pi/8) = (2 - sqrt2)/4`
`sin (pi/8) = +- (sqrt(2 - sqrt2)/2)`
Since `sin (pi/8)` is positive, take the positive value.
Finally:
`P = ((sqrt2)/2)(sqrt(2 - sqrt2)/2)`