Question

How do you express `sin(pi/ 6 ) * cos( ( pi) / 8 )` without using products of trigonometric functions?

Answer 1

`sqrt(2 + sqrt2)/4`

Explanation:

`P = sin (pi/6).cos (pi/8)`
Trig table gives --> sin (pi/6) = 1/2
`P = (1/2)cos (pi/8)`.
We can evaluate `cos (pi/8)` by using trig identity:
`2cos^2 a = 1 + cos 2a`
In this case:
`2cos^2 (pi/8) = 1 + cos (pi/4) = 1 + sqrt2/2 = (2 + sqrt2)/2`
`cos^2 (pi/8) = (2 + sqrt2)/4`
`cos (pi/8) = +- sqrt(2 + sqrt2)/2`.
Since `cos (pi/8)` is positive, take the positive answer.
Finally,
`P = (1/2)cos (pi/8) = sqrt(2 + sqrt2)/4`