Question

How do you express `cos(pi/ 3 ) * sin( ( 15 pi) / 8 )` without using products of trigonometric functions?

Answer 1

`P = 1/2 - (sqrt(2 + sqrt2)/2)`

Explanation:

`P = cos (pi/3).sin ((15pi)/8)`
`cos (pi/3) = 1/2`
`sin ((15pi)/8) = sin (2pi - pi/8) = - sin pi/8.`
Find `sin (pi/8)` by using trig identity:
`cos (pi/4) = 2cos^2 (pi/8) - 1 = sqrt2/2`
`2cos^2 (pi/8) = 1 + sqrt2/2 = (2 + sqrt2)/2`
`cos^2 (pi/8)^2 = (2 + sqrt2)/4`
`cos (pi/8) = (sqrt(2 + sqrt2))/2` (cos (pi/8) is positive.)
`P = 1/2 - (sqrt(2 + sqrt2))/2`