Question

How do you express `cos(pi/ 3 ) * sin( ( 9 pi) / 8 )` without using products of trigonometric functions?

Answer 1

`P = - (1/2)sin (pi/8)`

Explanation:

Product `P = cos (pi/3).sin ((9pi)/8)`
Trig table --> `cos (pi/3) = 1/2`
`sin ((9pi)/8) = sin (pi/8 + pi) = - sin (pi/8)`
P can be expressed as:
`P = - (1/2)sin (pi/8).`
Note. We can evaluate `sin (pi/8)` by using the trig identity:
`cos 2a = 1 - 2sin^2 a.`
`cos ((2pi)/8) = cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)`
`sin^2 (pi/8) = 1 - sqrt2 = (2 - sqrt2)/4`
`sin (pi/8) = sqrt(2 - sqrt2)/2`.
Take the positive value since sin (pi/8) is positive.
`P = - (1/4)sqrt(2 - sqrt2)`