Question

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

Answer 1

`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`